{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "# 5D Kerr-AdS spacetime with a Nambu-Goto string\n", "\n", "## Case a = b with global AdS coordinates" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "This [SageMath](https://www.sagemath.org/) notebook is relative to the article *Holographic drag force in 5d Kerr-AdS black hole* by Irina Ya. Aref'eva, Anastasia A. Golubtsova and Eric Gourgoulhon, [arXiv:2004.12984](https://arxiv.org/abs/2004.12984).\n", "\n", "The involved differential geometry computations are based on tools developed through the [SageManifolds](https://sagemanifolds.obspm.fr) project." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "*NB:* a version of SageMath at least equal to 8.2 is required to run this notebook:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/plain": [ "'SageMath version 9.3, Release Date: 2021-05-09'" ] }, "execution_count": 1, "metadata": { }, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "First we set up the notebook to display mathematical objects using LaTeX rendering:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "Since some computations are quite long, we ask for running them in parallel on 8 cores:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "Parallelism().set(nproc=1) # only nproc=1 works on CoCalc" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "## Spacetime manifold\n", "\n", "We declare the Kerr-AdS spacetime as a 5-dimensional Lorentzian manifold:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "5-dimensional Lorentzian manifold M\n" ] } ], "source": [ "M = Manifold(5, 'M', r'\\mathcal{M}', structure='Lorentzian', metric_name='G')\n", "print(M)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "Let us define **Boyer-Lindquist-type coordinates (rational polynomial version)** on $\\mathcal{M}$, via the method `chart()`, the argument of which is a string expressing the coordinates names, their ranges (the default is $(-\\infty,+\\infty)$) and their LaTeX symbols:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(t, r, {\\mu}, {\\phi}, {\\psi})\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(t, r, {\\mu}, {\\phi}, {\\psi})\\right)$$" ], "text/plain": [ "Chart (M, (t, r, mu, ph, ps))" ] }, "execution_count": 5, "metadata": { }, "output_type": "execute_result" } ], "source": [ "BL. = M.chart(r't r:(0,+oo) mu:(0,1):\\mu ph:(0,2*pi):\\phi ps:(0,2*pi):\\psi')\n", "BL" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "The coordinate $\\mu$ is related to the standard Boyer-Lindquist coordinate $\\theta$ by\n", "$$ \\mu = \\cos\\theta$$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The coordinate ranges are" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\mu} :\\ \\left( 0 , 1 \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right) ;\\quad {\\psi} :\\ \\left( 0 , 2 \\, \\pi \\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\mu} :\\ \\left( 0 , 1 \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right) ;\\quad {\\psi} :\\ \\left( 0 , 2 \\, \\pi \\right)$$" ], "text/plain": [ "t: (-oo, +oo); r: (0, +oo); mu: (0, 1); ph: (0, 2*pi); ps: (0, 2*pi)" ] }, "execution_count": 6, "metadata": { }, "output_type": "execute_result" } ], "source": [ "BL.coord_range()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Note that contrary to the 4-dimensional case, the range of $\\mu$ is $(0,1)$ only (cf. Sec. 1.2 of [R.C. Myers, arXiv:1111.1903](https://arxiv.org/abs/1111.1903) or Sec. 2 of [G.W. Gibbons, H. Lüb, Don N. Page, C.N. Pope, J. Geom. Phys. **53**, 49 (2005)](https://doi.org/10.1016/j.geomphys.2004.05.001)). In other words, the range of $\\theta$ is $\\left(0, \\frac{\\pi}{2}\\right)$ only. " ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "## Metric tensor\n", "\n", "The 4 parameters $m$, $a$, $b$ and $\\ell$ of the Kerr-AdS spacetime are declared as symbolic variables, $a$ and $b$ being the two angular momentum parameters and $\\ell$ being related to the cosmological constant by $\\Lambda = - 6 \\ell^2$:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(m, a, b\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(m, a, b\\right)$$" ], "text/plain": [ "(m, a, b)" ] }, "execution_count": 7, "metadata": { }, "output_type": "execute_result" } ], "source": [ "var('m a b', domain='real')" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\ell}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\ell}$$" ], "text/plain": [ "l" ] }, "execution_count": 8, "metadata": { }, "output_type": "execute_result" } ], "source": [ "var('l', domain='real', latex_name=r'\\ell')" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "# Particular cases\n", "# m = 0\n", "# a = 0\n", "# b = 0\n", "b = a" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "assume(a > 0)\n", "assume(1 - a^2*l^2 > 0)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "Some auxiliary functions:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "keep_Delta = False # change to False to provide explicit expression for Delta_r, Xi_a, etc..." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "sig = (1 + r^2*l^2)/r^2\n", "costh2 = mu^2\n", "sinth2 = 1 - mu^2\n", "rho2 = r^2 + a^2*mu^2 + b^2*sinth2\n", "if keep_Delta:\n", " Delta_r = var('Delta_r', latex_name=r'\\Delta_r', domain='real')\n", " Delta_th = var('Delta_th', latex_name=r'\\Delta_\\theta', domain='real')\n", " if a == b:\n", " Xi_a = var('Xi', latex_name=r'\\Xi', domain='real')\n", " Xi_b = Xi_a\n", " else:\n", " Xi_a = var('Xi_a', latex_name=r'\\Xi_a', domain='real')\n", " Xi_b = var('Xi_b', latex_name=r'\\Xi_b', domain='real')\n", " #Delta_th = 1 - a^2*l^2*mu^2 - b^2*l^2*sinth2\n", " Xi_a = 1 - a^2*l^2\n", " Xi_b = 1 - b^2*l^2\n", "else:\n", " Delta_r = (r^2+a^2)*(r^2+b^2)*sig - 2*m\n", " Delta_th = 1 - a^2*l^2*mu^2 - b^2*l^2*sinth2\n", " Xi_a = 1 - a^2*l^2\n", " Xi_b = 1 - b^2*l^2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "The metric is set by its components in the coordinate frame associated with the Boyer-Lindquist-type coordinates, which is the current manifold's default frame. These components can be deduced from\n", "Eq. (5.22) of the article [S.W. Hawking, C.J. Hunter & M.M. Taylor-Robinson, Phys. Rev. D **59**, 064005 (1999)](https://doi.org/10.1103/PhysRevD.59.064005) (the check of agreement with this equation is performed below):" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}G = \\left( -\\frac{a^{4} {\\ell}^{2} + {\\ell}^{2} r^{4} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2} - 2 \\, m}{a^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\psi} + \\left( \\frac{a^{2} r^{2} + r^{4}}{{\\ell}^{2} r^{6} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{4} + a^{4} + {\\left(a^{4} {\\ell}^{2} + 2 \\, a^{2} - 2 \\, m\\right)} r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( -\\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1} \\right) \\mathrm{d} {\\mu}\\otimes \\mathrm{d} {\\mu} + \\left( -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + \\left( -\\frac{a^{6} {\\ell}^{2} - 2 \\, a^{2} m {\\mu}^{4} + {\\left(a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} r^{4} - a^{4} - 2 \\, a^{2} m - {\\left(a^{6} {\\ell}^{2} - a^{4} - 4 \\, a^{2} m\\right)} {\\mu}^{2} + 2 \\, {\\left(a^{4} {\\ell}^{2} - {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} - a^{2}\\right)} r^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\psi} + \\left( -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\right) \\mathrm{d} {\\psi}\\otimes \\mathrm{d} t + \\left( -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\right) \\mathrm{d} {\\psi}\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} r^{4} - 2 \\, {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} r^{2} - {\\left(a^{6} {\\ell}^{2} - a^{4}\\right)} {\\mu}^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\right) \\mathrm{d} {\\psi}\\otimes \\mathrm{d} {\\psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}G = \\left( -\\frac{a^{4} {\\ell}^{2} + {\\ell}^{2} r^{4} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2} - 2 \\, m}{a^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\psi} + \\left( \\frac{a^{2} r^{2} + r^{4}}{{\\ell}^{2} r^{6} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{4} + a^{4} + {\\left(a^{4} {\\ell}^{2} + 2 \\, a^{2} - 2 \\, m\\right)} r^{2}} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( -\\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1} \\right) \\mathrm{d} {\\mu}\\otimes \\mathrm{d} {\\mu} + \\left( -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + \\left( -\\frac{a^{6} {\\ell}^{2} - 2 \\, a^{2} m {\\mu}^{4} + {\\left(a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} r^{4} - a^{4} - 2 \\, a^{2} m - {\\left(a^{6} {\\ell}^{2} - a^{4} - 4 \\, a^{2} m\\right)} {\\mu}^{2} + 2 \\, {\\left(a^{4} {\\ell}^{2} - {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} - a^{2}\\right)} r^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi} + \\left( -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\psi} + \\left( -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\right) \\mathrm{d} {\\psi}\\otimes \\mathrm{d} t + \\left( -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\right) \\mathrm{d} {\\psi}\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} r^{4} - 2 \\, {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} r^{2} - {\\left(a^{6} {\\ell}^{2} - a^{4}\\right)} {\\mu}^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\right) \\mathrm{d} {\\psi}\\otimes \\mathrm{d} {\\psi}$$" ], "text/plain": [ "G = -(a^4*l^2 + l^2*r^4 + (2*a^2*l^2 + 1)*r^2 + a^2 - 2*m)/(a^2 + r^2) dt*dt - (a^5*l^2 - (a*l^2*mu^2 - a*l^2)*r^4 - (a^5*l^2 - 2*a*m)*mu^2 - 2*(a^3*l^2*mu^2 - a^3*l^2)*r^2 - 2*a*m)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) dt*dph - (2*a^3*l^2*mu^2*r^2 + a*l^2*mu^2*r^4 + (a^5*l^2 - 2*a*m)*mu^2)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) dt*dps + (a^2*r^2 + r^4)/(l^2*r^6 + (2*a^2*l^2 + 1)*r^4 + a^4 + (a^4*l^2 + 2*a^2 - 2*m)*r^2) dr*dr - (a^2 + r^2)/(a^2*l^2 - (a^2*l^2 - 1)*mu^2 - 1) dmu*dmu - (a^5*l^2 - (a*l^2*mu^2 - a*l^2)*r^4 - (a^5*l^2 - 2*a*m)*mu^2 - 2*(a^3*l^2*mu^2 - a^3*l^2)*r^2 - 2*a*m)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) dph*dt - (a^6*l^2 - 2*a^2*m*mu^4 + (a^2*l^2 - (a^2*l^2 - 1)*mu^2 - 1)*r^4 - a^4 - 2*a^2*m - (a^6*l^2 - a^4 - 4*a^2*m)*mu^2 + 2*(a^4*l^2 - (a^4*l^2 - a^2)*mu^2 - a^2)*r^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) dph*dph - 2*(a^2*m*mu^4 - a^2*m*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) dph*dps - (2*a^3*l^2*mu^2*r^2 + a*l^2*mu^2*r^4 + (a^5*l^2 - 2*a*m)*mu^2)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) dps*dt - 2*(a^2*m*mu^4 - a^2*m*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) dps*dph + (2*a^2*m*mu^4 - (a^2*l^2 - 1)*mu^2*r^4 - 2*(a^4*l^2 - a^2)*mu^2*r^2 - (a^6*l^2 - a^4)*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) dps*dps" ] }, "execution_count": 13, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G = M.metric()\n", "tmp = 1/rho2*( -Delta_r + Delta_th*(a^2*sinth2 + b^2*mu^2) + a^2*b^2*sig )\n", "G[0,0] = tmp.simplify_full()\n", "tmp = a*sinth2/(rho2*Xi_a)*( Delta_r - (r^2+a^2)*(Delta_th + b^2*sig) )\n", "G[0,3] = tmp.simplify_full()\n", "tmp = b*mu^2/(rho2*Xi_b)*( Delta_r - (r^2+b^2)*(Delta_th + a^2*sig) )\n", "G[0,4] = tmp.simplify_full()\n", "G[1,1] = (rho2/Delta_r).simplify_full()\n", "G[2,2] = (rho2/Delta_th/(1-mu^2)).simplify_full()\n", "tmp = sinth2/(rho2*Xi_a^2)*( - Delta_r*a^2*sinth2 + (r^2+a^2)^2*(Delta_th + sig*b^2*sinth2) ) \n", "G[3,3] = tmp.simplify_full()\n", "tmp = a*b*sinth2*mu^2/(rho2*Xi_a*Xi_b)*( - Delta_r + sig*(r^2+a^2)*(r^2+b^2) )\n", "G[3,4] = tmp.simplify_full()\n", "tmp = mu^2/(rho2*Xi_b^2)*( - Delta_r*b^2*mu^2 + (r^2+b^2)^2*(Delta_th + sig*a^2*mu^2) )\n", "G[4,4] = tmp.simplify_full()\n", "G.display()" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & -\\frac{a^{4} {\\ell}^{2} + {\\ell}^{2} r^{4} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2} - 2 \\, m}{a^{2} + r^{2}} \\\\ G_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ G_{ \\, t \\, {\\psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\psi}} } & = & -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ G_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & \\frac{a^{2} r^{2} + r^{4}}{{\\ell}^{2} r^{6} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{4} + a^{4} + {\\left(a^{4} {\\ell}^{2} + 2 \\, a^{2} - 2 \\, m\\right)} r^{2}} \\\\ G_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & -\\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1} \\\\ G_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & -\\frac{a^{6} {\\ell}^{2} - 2 \\, a^{2} m {\\mu}^{4} + {\\left(a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} r^{4} - a^{4} - 2 \\, a^{2} m - {\\left(a^{6} {\\ell}^{2} - a^{4} - 4 \\, a^{2} m\\right)} {\\mu}^{2} + 2 \\, {\\left(a^{4} {\\ell}^{2} - {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} - a^{2}\\right)} r^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ G_{ \\, {\\phi} \\, {\\psi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\psi}} } & = & -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ G_{ \\, {\\psi} \\, {\\psi} }^{ \\phantom{\\, {\\psi}}\\phantom{\\, {\\psi}} } & = & \\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} r^{4} - 2 \\, {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} r^{2} - {\\left(a^{6} {\\ell}^{2} - a^{4}\\right)} {\\mu}^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & -\\frac{a^{4} {\\ell}^{2} + {\\ell}^{2} r^{4} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2} - 2 \\, m}{a^{2} + r^{2}} \\\\ G_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ G_{ \\, t \\, {\\psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\psi}} } & = & -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ G_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & \\frac{a^{2} r^{2} + r^{4}}{{\\ell}^{2} r^{6} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{4} + a^{4} + {\\left(a^{4} {\\ell}^{2} + 2 \\, a^{2} - 2 \\, m\\right)} r^{2}} \\\\ G_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & -\\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1} \\\\ G_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & -\\frac{a^{6} {\\ell}^{2} - 2 \\, a^{2} m {\\mu}^{4} + {\\left(a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} r^{4} - a^{4} - 2 \\, a^{2} m - {\\left(a^{6} {\\ell}^{2} - a^{4} - 4 \\, a^{2} m\\right)} {\\mu}^{2} + 2 \\, {\\left(a^{4} {\\ell}^{2} - {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} - a^{2}\\right)} r^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ G_{ \\, {\\phi} \\, {\\psi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\psi}} } & = & -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ G_{ \\, {\\psi} \\, {\\psi} }^{ \\phantom{\\, {\\psi}}\\phantom{\\, {\\psi}} } & = & \\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} r^{4} - 2 \\, {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} r^{2} - {\\left(a^{6} {\\ell}^{2} - a^{4}\\right)} {\\mu}^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\end{array}$$" ], "text/plain": [ "G_t,t = -(a^4*l^2 + l^2*r^4 + (2*a^2*l^2 + 1)*r^2 + a^2 - 2*m)/(a^2 + r^2) \n", "G_t,ph = -(a^5*l^2 - (a*l^2*mu^2 - a*l^2)*r^4 - (a^5*l^2 - 2*a*m)*mu^2 - 2*(a^3*l^2*mu^2 - a^3*l^2)*r^2 - 2*a*m)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) \n", "G_t,ps = -(2*a^3*l^2*mu^2*r^2 + a*l^2*mu^2*r^4 + (a^5*l^2 - 2*a*m)*mu^2)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) \n", "G_r,r = (a^2*r^2 + r^4)/(l^2*r^6 + (2*a^2*l^2 + 1)*r^4 + a^4 + (a^4*l^2 + 2*a^2 - 2*m)*r^2) \n", "G_mu,mu = -(a^2 + r^2)/(a^2*l^2 - (a^2*l^2 - 1)*mu^2 - 1) \n", "G_ph,ph = -(a^6*l^2 - 2*a^2*m*mu^4 + (a^2*l^2 - (a^2*l^2 - 1)*mu^2 - 1)*r^4 - a^4 - 2*a^2*m - (a^6*l^2 - a^4 - 4*a^2*m)*mu^2 + 2*(a^4*l^2 - (a^4*l^2 - a^2)*mu^2 - a^2)*r^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) \n", "G_ph,ps = -2*(a^2*m*mu^4 - a^2*m*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) \n", "G_ps,ps = (2*a^2*m*mu^4 - (a^2*l^2 - 1)*mu^2*r^4 - 2*(a^4*l^2 - a^2)*mu^2*r^2 - (a^6*l^2 - a^4)*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) " ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G.display_comp(only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Check of agreement with Eq. (5.22) of Hawking et al or Eq. (2.3) of o\n", "\n", "We need the 1-forms $\\mathrm{d}t$, $\\mathrm{d}r$, $\\mathrm{d}\\mu$, $\\mathrm{d}\\phi$ and $\\mathrm{d}\\psi$:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathrm{d} t, \\mathrm{d} r, \\mathrm{d} {\\mu}, \\mathrm{d} {\\phi}, \\mathrm{d} {\\psi}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathrm{d} t, \\mathrm{d} r, \\mathrm{d} {\\mu}, \\mathrm{d} {\\phi}, \\mathrm{d} {\\psi}\\right)$$" ], "text/plain": [ "(1-form dt on the 5-dimensional Lorentzian manifold M,\n", " 1-form dr on the 5-dimensional Lorentzian manifold M,\n", " 1-form dmu on the 5-dimensional Lorentzian manifold M,\n", " 1-form dph on the 5-dimensional Lorentzian manifold M,\n", " 1-form dps on the 5-dimensional Lorentzian manifold M)" ] }, "execution_count": 15, "metadata": { }, "output_type": "execute_result" } ], "source": [ "dt, dr, dmu, dph, dps = (BL.coframe()[i] for i in M.irange())\n", "dt, dr, dmu, dph, dps" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1-form dt on the 5-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(dt)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "In agreement with $\\mu = \\cos\\theta$, we introduce the 1-form $\\mathrm{d}\\theta = - \\mathrm{d}\\mu /\\sin\\theta $, with \n", "$\\sin\\theta = \\sqrt{1-\\mu^2}$ since $\\theta\\in\\left(0, \\frac{\\pi}{2}\\right)$:" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "dth = - 1/sqrt(1 - mu^2)*dmu" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d} t + \\left( -\\frac{a {\\mu}^{2} - a}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\phi} + \\left( \\frac{a {\\mu}^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d} t + \\left( -\\frac{a {\\mu}^{2} - a}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\phi} + \\left( \\frac{a {\\mu}^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\psi}$$" ], "text/plain": [ "dt - (a*mu^2 - a)/(a^2*l^2 - 1) dph + a*mu^2/(a^2*l^2 - 1) dps" ] }, "execution_count": 18, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = dt - a*sinth2/Xi_a*dph - b*costh2/Xi_b*dps\n", "s1.display()" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}a \\mathrm{d} t + \\left( \\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\phi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}a \\mathrm{d} t + \\left( \\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\phi}$$" ], "text/plain": [ "a dt + (a^2 + r^2)/(a^2*l^2 - 1) dph" ] }, "execution_count": 19, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = a*dt - (r^2 + a^2)/Xi_a*dph\n", "s2.display()" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}a \\mathrm{d} t + \\left( \\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}a \\mathrm{d} t + \\left( \\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\psi}$$" ], "text/plain": [ "a dt + (a^2 + r^2)/(a^2*l^2 - 1) dps" ] }, "execution_count": 20, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s3 = b*dt - (r^2 + b^2)/Xi_b*dps\n", "s3.display()" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}a^{2} \\mathrm{d} t + \\left( -\\frac{a^{3} {\\mu}^{2} - a^{3} + {\\left(a {\\mu}^{2} - a\\right)} r^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\phi} + \\left( \\frac{a^{3} {\\mu}^{2} + a {\\mu}^{2} r^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}a^{2} \\mathrm{d} t + \\left( -\\frac{a^{3} {\\mu}^{2} - a^{3} + {\\left(a {\\mu}^{2} - a\\right)} r^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\phi} + \\left( \\frac{a^{3} {\\mu}^{2} + a {\\mu}^{2} r^{2}}{a^{2} {\\ell}^{2} - 1} \\right) \\mathrm{d} {\\psi}$$" ], "text/plain": [ "a^2 dt - (a^3*mu^2 - a^3 + (a*mu^2 - a)*r^2)/(a^2*l^2 - 1) dph + (a^3*mu^2 + a*mu^2*r^2)/(a^2*l^2 - 1) dps" ] }, "execution_count": 21, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s4 = a*b*dt - b*(r^2 + a^2)*sinth2/Xi_a * dph - a*(r^2 + b^2)*costh2/Xi_b * dps\n", "s4.display()" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} X_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & -\\frac{a^{4} {\\ell}^{2} + {\\ell}^{2} r^{4} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2} - 2 \\, m}{a^{2} + r^{2}} \\\\ X_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ X_{ \\, t \\, {\\psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\psi}} } & = & -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ X_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & -\\frac{a^{2} {\\mu}^{2} - {\\left({\\mu}^{2} - 1\\right)} a^{2} + r^{2}}{2 \\, m - \\frac{{\\left({\\ell}^{2} r^{2} + 1\\right)} {\\left(a^{2} + r^{2}\\right)}^{2}}{r^{2}}} \\\\ X_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & -\\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1} \\\\ X_{ \\, {\\phi} \\, t }^{ \\phantom{\\, {\\phi}}\\phantom{\\, t} } & = & -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ X_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & -\\frac{a^{6} {\\ell}^{2} - 2 \\, a^{2} m {\\mu}^{4} + {\\left(a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} r^{4} - a^{4} - 2 \\, a^{2} m - {\\left(a^{6} {\\ell}^{2} - a^{4} - 4 \\, a^{2} m\\right)} {\\mu}^{2} + 2 \\, {\\left(a^{4} {\\ell}^{2} - {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} - a^{2}\\right)} r^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ X_{ \\, {\\phi} \\, {\\psi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\psi}} } & = & -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ X_{ \\, {\\psi} \\, t }^{ \\phantom{\\, {\\psi}}\\phantom{\\, t} } & = & -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ X_{ \\, {\\psi} \\, {\\phi} }^{ \\phantom{\\, {\\psi}}\\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ X_{ \\, {\\psi} \\, {\\psi} }^{ \\phantom{\\, {\\psi}}\\phantom{\\, {\\psi}} } & = & \\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} r^{4} - 2 \\, {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} r^{2} - {\\left(a^{6} {\\ell}^{2} - a^{4}\\right)} {\\mu}^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} X_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & -\\frac{a^{4} {\\ell}^{2} + {\\ell}^{2} r^{4} + {\\left(2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2} - 2 \\, m}{a^{2} + r^{2}} \\\\ X_{ \\, t \\, {\\phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\phi}} } & = & -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ X_{ \\, t \\, {\\psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\psi}} } & = & -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ X_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & -\\frac{a^{2} {\\mu}^{2} - {\\left({\\mu}^{2} - 1\\right)} a^{2} + r^{2}}{2 \\, m - \\frac{{\\left({\\ell}^{2} r^{2} + 1\\right)} {\\left(a^{2} + r^{2}\\right)}^{2}}{r^{2}}} \\\\ X_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & -\\frac{a^{2} + r^{2}}{a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1} \\\\ X_{ \\, {\\phi} \\, t }^{ \\phantom{\\, {\\phi}}\\phantom{\\, t} } & = & -\\frac{a^{5} {\\ell}^{2} - {\\left(a {\\ell}^{2} {\\mu}^{2} - a {\\ell}^{2}\\right)} r^{4} - {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2} - 2 \\, {\\left(a^{3} {\\ell}^{2} {\\mu}^{2} - a^{3} {\\ell}^{2}\\right)} r^{2} - 2 \\, a m}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ X_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & -\\frac{a^{6} {\\ell}^{2} - 2 \\, a^{2} m {\\mu}^{4} + {\\left(a^{2} {\\ell}^{2} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} r^{4} - a^{4} - 2 \\, a^{2} m - {\\left(a^{6} {\\ell}^{2} - a^{4} - 4 \\, a^{2} m\\right)} {\\mu}^{2} + 2 \\, {\\left(a^{4} {\\ell}^{2} - {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} - a^{2}\\right)} r^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ X_{ \\, {\\phi} \\, {\\psi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\psi}} } & = & -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ X_{ \\, {\\psi} \\, t }^{ \\phantom{\\, {\\psi}}\\phantom{\\, t} } & = & -\\frac{2 \\, a^{3} {\\ell}^{2} {\\mu}^{2} r^{2} + a {\\ell}^{2} {\\mu}^{2} r^{4} + {\\left(a^{5} {\\ell}^{2} - 2 \\, a m\\right)} {\\mu}^{2}}{a^{4} {\\ell}^{2} + {\\left(a^{2} {\\ell}^{2} - 1\\right)} r^{2} - a^{2}} \\\\ X_{ \\, {\\psi} \\, {\\phi} }^{ \\phantom{\\, {\\psi}}\\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\\\ X_{ \\, {\\psi} \\, {\\psi} }^{ \\phantom{\\, {\\psi}}\\phantom{\\, {\\psi}} } & = & \\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} r^{4} - 2 \\, {\\left(a^{4} {\\ell}^{2} - a^{2}\\right)} {\\mu}^{2} r^{2} - {\\left(a^{6} {\\ell}^{2} - a^{4}\\right)} {\\mu}^{2}}{a^{6} {\\ell}^{4} - 2 \\, a^{4} {\\ell}^{2} + {\\left(a^{4} {\\ell}^{4} - 2 \\, a^{2} {\\ell}^{2} + 1\\right)} r^{2} + a^{2}} \\end{array}$$" ], "text/plain": [ "X_t,t = -(a^4*l^2 + l^2*r^4 + (2*a^2*l^2 + 1)*r^2 + a^2 - 2*m)/(a^2 + r^2) \n", "X_t,ph = -(a^5*l^2 - (a*l^2*mu^2 - a*l^2)*r^4 - (a^5*l^2 - 2*a*m)*mu^2 - 2*(a^3*l^2*mu^2 - a^3*l^2)*r^2 - 2*a*m)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) \n", "X_t,ps = -(2*a^3*l^2*mu^2*r^2 + a*l^2*mu^2*r^4 + (a^5*l^2 - 2*a*m)*mu^2)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) \n", "X_r,r = -(a^2*mu^2 - (mu^2 - 1)*a^2 + r^2)/(2*m - (l^2*r^2 + 1)*(a^2 + r^2)^2/r^2) \n", "X_mu,mu = -(a^2 + r^2)/(a^2*l^2 - (a^2*l^2 - 1)*mu^2 - 1) \n", "X_ph,t = -(a^5*l^2 - (a*l^2*mu^2 - a*l^2)*r^4 - (a^5*l^2 - 2*a*m)*mu^2 - 2*(a^3*l^2*mu^2 - a^3*l^2)*r^2 - 2*a*m)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) \n", "X_ph,ph = -(a^6*l^2 - 2*a^2*m*mu^4 + (a^2*l^2 - (a^2*l^2 - 1)*mu^2 - 1)*r^4 - a^4 - 2*a^2*m - (a^6*l^2 - a^4 - 4*a^2*m)*mu^2 + 2*(a^4*l^2 - (a^4*l^2 - a^2)*mu^2 - a^2)*r^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) \n", "X_ph,ps = -2*(a^2*m*mu^4 - a^2*m*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) \n", "X_ps,t = -(2*a^3*l^2*mu^2*r^2 + a*l^2*mu^2*r^4 + (a^5*l^2 - 2*a*m)*mu^2)/(a^4*l^2 + (a^2*l^2 - 1)*r^2 - a^2) \n", "X_ps,ph = -2*(a^2*m*mu^4 - a^2*m*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) \n", "X_ps,ps = (2*a^2*m*mu^4 - (a^2*l^2 - 1)*mu^2*r^4 - 2*(a^4*l^2 - a^2)*mu^2*r^2 - (a^6*l^2 - a^4)*mu^2)/(a^6*l^4 - 2*a^4*l^2 + (a^4*l^4 - 2*a^2*l^2 + 1)*r^2 + a^2) " ] }, "execution_count": 22, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G0 = - Delta_r/rho2 * s1*s1 + Delta_th*sinth2/rho2 * s2*s2 + Delta_th*costh2/rho2 * s3*s3 \\\n", " + rho2/Delta_r * dr*dr + rho2/Delta_th * dth*dth + sig/rho2 * s4*s4\n", "G0.display_comp(only_nonredundant=True)" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 23, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G0 == G" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "## Einstein equation\n", "\n", "The Ricci tensor of $g$ is" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "if not keep_Delta:\n", " # Ric = G.ricci()\n", " # print(Ric)\n", " pass" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "if not keep_Delta:\n", " # show(Ric.display_comp(only_nonredundant=True))\n", " pass" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "Let us check that $g$ is a solution of the vacuum Einstein equation with the cosmological constant $\\Lambda = - 6 \\ell^2$:" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "Lambda = -6*l^2\n", "if not keep_Delta:\n", " # print(Ric == 2/3*Lambda*G)\n", " pass" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Check of Eq. (2.10)\n", "\n", "One must have $a=b$ and `keep_Delta == False` for the test to pass:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "True\n" ] } ], "source": [ "if a == b and not keep_Delta:\n", " G1 = - (1 + rho2*l^2 - 2*m/rho2) * dt*dt + rho2/Delta_r * dr*dr \\\n", " + rho2/Delta_th * dth*dth \\\n", " + sinth2/Xi_a^2*(rho2*Xi_a + 2*a^2*m/rho2*sinth2) * dph * dph \\\n", " + costh2/Xi_a^2*(rho2*Xi_a + 2*a^2*m/rho2*costh2) * dps * dps \\\n", " + a*sinth2/Xi_a*(rho2*l^2 - 2*m/rho2) * (dt*dph + dph*dt) \\\n", " + a*costh2/Xi_a*(rho2*l^2 - 2*m/rho2) * (dt*dps + dps*dt) \\\n", " + 2*m*a^2*sinth2*costh2/Xi_a^2/rho2 * (dph*dps + dps*dph)\n", " print(G1 == G)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Global AdS coordinates" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(t, y, {\\mu}, {\\Phi}, {\\Psi})\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(t, y, {\\mu}, {\\Phi}, {\\Psi})\\right)$$" ], "text/plain": [ "Chart (M, (t, y, mu, Ph, Ps))" ] }, "execution_count": 28, "metadata": { }, "output_type": "execute_result" } ], "source": [ "ADS. = M.chart(r't y:(a/sqrt(1-a^2*l^2),+oo) mu:(0,1):\\mu Ph:(0,2*pi):\\Phi Ps:(0,2*pi):\\Psi')\n", "ADS" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad y :\\ \\left( \\frac{a}{\\sqrt{-a^{2} {\\ell}^{2} + 1}} , +\\infty \\right) ;\\quad {\\mu} :\\ \\left( 0 , 1 \\right) ;\\quad {\\Phi} :\\ \\left( 0 , 2 \\, \\pi \\right) ;\\quad {\\Psi} :\\ \\left( 0 , 2 \\, \\pi \\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad y :\\ \\left( \\frac{a}{\\sqrt{-a^{2} {\\ell}^{2} + 1}} , +\\infty \\right) ;\\quad {\\mu} :\\ \\left( 0 , 1 \\right) ;\\quad {\\Phi} :\\ \\left( 0 , 2 \\, \\pi \\right) ;\\quad {\\Psi} :\\ \\left( 0 , 2 \\, \\pi \\right)$$" ], "text/plain": [ "t: (-oo, +oo); y: (a/sqrt(-a^2*l^2 + 1), +oo); mu: (0, 1); Ph: (0, 2*pi); Ps: (0, 2*pi)" ] }, "execution_count": 29, "metadata": { }, "output_type": "execute_result" } ], "source": [ "ADS.coord_range()" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\verb|t|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|r|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, r > 0, \\verb|mu|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\mu} > 0, {\\mu} < 1, \\verb|ph|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\phi} > 0, {\\phi} < 2 \\, \\pi, \\verb|ps|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\psi} > 0, {\\psi} < 2 \\, \\pi, \\verb|m|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|a|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|b|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|l|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, a > 0, -a^{2} {\\ell}^{2} + 1 > 0, \\verb|y|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, y > \\frac{a}{\\sqrt{-a^{2} {\\ell}^{2} + 1}}, \\verb|Ph|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\Phi} > 0, {\\Phi} < 2 \\, \\pi, \\verb|Ps|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\Psi} > 0, {\\Psi} < 2 \\, \\pi\\right]$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\verb|t|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|r|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, r > 0, \\verb|mu|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\mu} > 0, {\\mu} < 1, \\verb|ph|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\phi} > 0, {\\phi} < 2 \\, \\pi, \\verb|ps|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\psi} > 0, {\\psi} < 2 \\, \\pi, \\verb|m|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|a|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|b|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|l|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, a > 0, -a^{2} {\\ell}^{2} + 1 > 0, \\verb|y|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, y > \\frac{a}{\\sqrt{-a^{2} {\\ell}^{2} + 1}}, \\verb|Ph|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\Phi} > 0, {\\Phi} < 2 \\, \\pi, \\verb|Ps|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\Psi} > 0, {\\Psi} < 2 \\, \\pi\\right]$$" ], "text/plain": [ "[t is real,\n", " r is real,\n", " r > 0,\n", " mu is real,\n", " mu > 0,\n", " mu < 1,\n", " ph is real,\n", " ph > 0,\n", " ph < 2*pi,\n", " ps is real,\n", " ps > 0,\n", " ps < 2*pi,\n", " m is real,\n", " a is real,\n", " b is real,\n", " l is real,\n", " a > 0,\n", " -a^2*l^2 + 1 > 0,\n", " y is real,\n", " y > a/sqrt(-a^2*l^2 + 1),\n", " Ph is real,\n", " Ph > 0,\n", " Ph < 2*pi,\n", " Ps is real,\n", " Ps > 0,\n", " Ps < 2*pi]" ] }, "execution_count": 30, "metadata": { }, "output_type": "execute_result" } ], "source": [ "assumptions()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Transition from the Boyer-Lindquist coordinates to the AdS global coordinates, according to Eq. (5.24) of [S.W. Hawking, C.J. Hunter & M.M. Taylor-Robinson, Phys. Rev. D **59**, 064005 (1999)](https://doi.org/10.1103/PhysRevD.59.064005):" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & t \\\\ y & = & \\frac{\\sqrt{a^{2} + r^{2}}}{\\sqrt{-a^{2} {\\ell}^{2} + 1}} \\\\ {\\mu} & = & {\\mu} \\\\ {\\Phi} & = & a {\\ell}^{2} t + {\\phi} \\\\ {\\Psi} & = & a {\\ell}^{2} t + {\\psi} \\end{array}\\right.$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & t \\\\ y & = & \\frac{\\sqrt{a^{2} + r^{2}}}{\\sqrt{-a^{2} {\\ell}^{2} + 1}} \\\\ {\\mu} & = & {\\mu} \\\\ {\\Phi} & = & a {\\ell}^{2} t + {\\phi} \\\\ {\\Psi} & = & a {\\ell}^{2} t + {\\psi} \\end{array}\\right.$$" ], "text/plain": [ "t = t\n", "y = sqrt(a^2 + r^2)/sqrt(-a^2*l^2 + 1)\n", "mu = mu\n", "Ph = a*l^2*t + ph\n", "Ps = a*l^2*t + ps" ] }, "execution_count": 31, "metadata": { }, "output_type": "execute_result" } ], "source": [ "BL_to_ADS = BL.transition_map(ADS, [t, sqrt(r^2 + a^2)/sqrt(Xi_a), mu, \n", " ph + a*l^2*t, ps + a*l^2*t])\n", "BL_to_ADS.display()" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Check of the inverse coordinate transformation:\n", " t == t *passed*\n", " r == r *passed*\n", " mu == mu *passed*\n", " ph == ph *passed*\n", " ps == ps *passed*\n", " t == t *passed*\n", " y == abs(y) **failed**\n", " mu == mu *passed*\n", " Ph == Ph *passed*\n", " Ps == Ps *passed*\n", "NB: a failed report can reflect a mere lack of simplification.\n" ] }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & t \\\\ r & = & \\sqrt{-{\\left(a^{2} {\\ell}^{2} - 1\\right)} y^{2} - a^{2}} \\\\ {\\mu} & = & {\\mu} \\\\ {\\phi} & = & -a {\\ell}^{2} t + {\\Phi} \\\\ {\\psi} & = & -a {\\ell}^{2} t + {\\Psi} \\end{array}\\right.$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} t & = & t \\\\ r & = & \\sqrt{-{\\left(a^{2} {\\ell}^{2} - 1\\right)} y^{2} - a^{2}} \\\\ {\\mu} & = & {\\mu} \\\\ {\\phi} & = & -a {\\ell}^{2} t + {\\Phi} \\\\ {\\psi} & = & -a {\\ell}^{2} t + {\\Psi} \\end{array}\\right.$$" ], "text/plain": [ "t = t\n", "r = sqrt(-(a^2*l^2 - 1)*y^2 - a^2)\n", "mu = mu\n", "ph = -a*l^2*t + Ph\n", "ps = -a*l^2*t + Ps" ] }, "execution_count": 32, "metadata": { }, "output_type": "execute_result" } ], "source": [ "BL_to_ADS.set_inverse(t, sqrt(Xi_a*y^2 - a^2), mu, Ph - a*l^2*t, Ps - a*l^2*t, \n", " verbose=True)\n", "BL_to_ADS.inverse().display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Metric tensor is global AdS coordinates" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & -\\frac{{\\left(a^{6} {\\ell}^{8} - 3 \\, a^{4} {\\ell}^{6} + 3 \\, a^{2} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{4} + {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2} + 2 \\, m}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, t \\, {\\Phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, {\\left(a m {\\mu}^{2} - a m\\right)}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, t \\, {\\Psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\Psi}} } & = & \\frac{2 \\, a m {\\mu}^{2}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & \\frac{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{4}}{{\\left(a^{6} {\\ell}^{8} - 3 \\, a^{4} {\\ell}^{6} + 3 \\, a^{2} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{6} + {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{4} - 2 \\, {\\left(a^{2} {\\ell}^{2} - 1\\right)} m y^{2} - 2 \\, a^{2} m} \\\\ G_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & -\\frac{y^{2}}{{\\mu}^{2} - 1} \\\\ G_{ \\, {\\Phi} \\, {\\Phi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, a^{2} m {\\mu}^{4} - 4 \\, a^{2} m {\\mu}^{2} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} y^{4} + 2 \\, a^{2} m}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, {\\Phi} \\, {\\Psi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Psi}} } & = & \\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, {\\Psi} \\, {\\Psi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Psi}} } & = & -\\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} y^{4}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & -\\frac{{\\left(a^{6} {\\ell}^{8} - 3 \\, a^{4} {\\ell}^{6} + 3 \\, a^{2} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{4} + {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2} + 2 \\, m}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, t \\, {\\Phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, {\\left(a m {\\mu}^{2} - a m\\right)}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, t \\, {\\Psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\Psi}} } & = & \\frac{2 \\, a m {\\mu}^{2}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & \\frac{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{4}}{{\\left(a^{6} {\\ell}^{8} - 3 \\, a^{4} {\\ell}^{6} + 3 \\, a^{2} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{6} + {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{4} - 2 \\, {\\left(a^{2} {\\ell}^{2} - 1\\right)} m y^{2} - 2 \\, a^{2} m} \\\\ G_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & -\\frac{y^{2}}{{\\mu}^{2} - 1} \\\\ G_{ \\, {\\Phi} \\, {\\Phi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, a^{2} m {\\mu}^{4} - 4 \\, a^{2} m {\\mu}^{2} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} y^{4} + 2 \\, a^{2} m}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, {\\Phi} \\, {\\Psi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Psi}} } & = & \\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, {\\Psi} \\, {\\Psi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Psi}} } & = & -\\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} y^{4}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\end{array}$$" ], "text/plain": [ "G_t,t = -((a^6*l^8 - 3*a^4*l^6 + 3*a^2*l^4 - l^2)*y^4 + (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2 + 2*m)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n", "G_t,Ph = -2*(a*m*mu^2 - a*m)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n", "G_t,Ps = 2*a*m*mu^2/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n", "G_y,y = (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^4/((a^6*l^8 - 3*a^4*l^6 + 3*a^2*l^4 - l^2)*y^6 + (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^4 - 2*(a^2*l^2 - 1)*m*y^2 - 2*a^2*m) \n", "G_mu,mu = -y^2/(mu^2 - 1) \n", "G_Ph,Ph = -(2*a^2*m*mu^4 - 4*a^2*m*mu^2 - (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*mu^2 - 1)*y^4 + 2*a^2*m)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n", "G_Ph,Ps = 2*(a^2*m*mu^4 - a^2*m*mu^2)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n", "G_Ps,Ps = -(2*a^2*m*mu^4 - (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*mu^2*y^4)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) " ] }, "execution_count": 33, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G.display_comp(chart=ADS, only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "From now on, we set the AdS coordinates as the default chart on $\\mathcal{M}$: " ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "M.set_default_chart(ADS)\n", "M.set_default_frame(ADS.frame())" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Then" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & -\\frac{{\\left(a^{6} {\\ell}^{8} - 3 \\, a^{4} {\\ell}^{6} + 3 \\, a^{2} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{4} + {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2} + 2 \\, m}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, t \\, {\\Phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, {\\left(a m {\\mu}^{2} - a m\\right)}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, t \\, {\\Psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\Psi}} } & = & \\frac{2 \\, a m {\\mu}^{2}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & \\frac{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{4}}{{\\left(a^{6} {\\ell}^{8} - 3 \\, a^{4} {\\ell}^{6} + 3 \\, a^{2} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{6} + {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{4} - 2 \\, {\\left(a^{2} {\\ell}^{2} - 1\\right)} m y^{2} - 2 \\, a^{2} m} \\\\ G_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & -\\frac{y^{2}}{{\\mu}^{2} - 1} \\\\ G_{ \\, {\\Phi} \\, {\\Phi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, a^{2} m {\\mu}^{4} - 4 \\, a^{2} m {\\mu}^{2} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} y^{4} + 2 \\, a^{2} m}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, {\\Phi} \\, {\\Psi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Psi}} } & = & \\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, {\\Psi} \\, {\\Psi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Psi}} } & = & -\\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} y^{4}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} G_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & -\\frac{{\\left(a^{6} {\\ell}^{8} - 3 \\, a^{4} {\\ell}^{6} + 3 \\, a^{2} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{4} + {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2} + 2 \\, m}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, t \\, {\\Phi} }^{ \\phantom{\\, t}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, {\\left(a m {\\mu}^{2} - a m\\right)}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, t \\, {\\Psi} }^{ \\phantom{\\, t}\\phantom{\\, {\\Psi}} } & = & \\frac{2 \\, a m {\\mu}^{2}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, y \\, y }^{ \\phantom{\\, y}\\phantom{\\, y} } & = & \\frac{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{4}}{{\\left(a^{6} {\\ell}^{8} - 3 \\, a^{4} {\\ell}^{6} + 3 \\, a^{2} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{6} + {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{4} - 2 \\, {\\left(a^{2} {\\ell}^{2} - 1\\right)} m y^{2} - 2 \\, a^{2} m} \\\\ G_{ \\, {\\mu} \\, {\\mu} }^{ \\phantom{\\, {\\mu}}\\phantom{\\, {\\mu}} } & = & -\\frac{y^{2}}{{\\mu}^{2} - 1} \\\\ G_{ \\, {\\Phi} \\, {\\Phi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Phi}} } & = & -\\frac{2 \\, a^{2} m {\\mu}^{4} - 4 \\, a^{2} m {\\mu}^{2} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} - 1\\right)} y^{4} + 2 \\, a^{2} m}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, {\\Phi} \\, {\\Psi} }^{ \\phantom{\\, {\\Phi}}\\phantom{\\, {\\Psi}} } & = & \\frac{2 \\, {\\left(a^{2} m {\\mu}^{4} - a^{2} m {\\mu}^{2}\\right)}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\\\ G_{ \\, {\\Psi} \\, {\\Psi} }^{ \\phantom{\\, {\\Psi}}\\phantom{\\, {\\Psi}} } & = & -\\frac{2 \\, a^{2} m {\\mu}^{4} - {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} {\\mu}^{2} y^{4}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}} \\end{array}$$" ], "text/plain": [ "G_t,t = -((a^6*l^8 - 3*a^4*l^6 + 3*a^2*l^4 - l^2)*y^4 + (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2 + 2*m)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n", "G_t,Ph = -2*(a*m*mu^2 - a*m)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n", "G_t,Ps = 2*a*m*mu^2/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n", "G_y,y = (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^4/((a^6*l^8 - 3*a^4*l^6 + 3*a^2*l^4 - l^2)*y^6 + (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^4 - 2*(a^2*l^2 - 1)*m*y^2 - 2*a^2*m) \n", "G_mu,mu = -y^2/(mu^2 - 1) \n", "G_Ph,Ph = -(2*a^2*m*mu^4 - 4*a^2*m*mu^2 - (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*mu^2 - 1)*y^4 + 2*a^2*m)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n", "G_Ph,Ps = 2*(a^2*m*mu^4 - a^2*m*mu^2)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) \n", "G_Ps,Ps = -(2*a^2*m*mu^4 - (a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*mu^2*y^4)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2) " ] }, "execution_count": 35, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G.display_comp(only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Comparison with Eq. (5.32) of [S.W. Hawking, C.J. Hunter & M.M. Taylor-Robinson, Phys. Rev. D **59**, 064005 (1999)](https://doi.org/10.1103/PhysRevD.59.064005) (or Eq. (2.18) of our paper):" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathrm{d} t, \\mathrm{d} y, \\mathrm{d} {\\mu}, \\mathrm{d} {\\Phi}, \\mathrm{d} {\\Psi}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathrm{d} t, \\mathrm{d} y, \\mathrm{d} {\\mu}, \\mathrm{d} {\\Phi}, \\mathrm{d} {\\Psi}\\right)$$" ], "text/plain": [ "(1-form dt on the 5-dimensional Lorentzian manifold M,\n", " 1-form dy on the 5-dimensional Lorentzian manifold M,\n", " 1-form dmu on the 5-dimensional Lorentzian manifold M,\n", " 1-form dPh on the 5-dimensional Lorentzian manifold M,\n", " 1-form dPs on the 5-dimensional Lorentzian manifold M)" ] }, "execution_count": 36, "metadata": { }, "output_type": "execute_result" } ], "source": [ "dt, dy, dmu, dPh, dPs = (ADS.coframe()[i] for i in M.irange())\n", "dt, dy, dmu, dPh, dPs" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d} t + \\left( a {\\mu}^{2} - a \\right) \\mathrm{d} {\\Phi} -a {\\mu}^{2} \\mathrm{d} {\\Psi}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d} t + \\left( a {\\mu}^{2} - a \\right) \\mathrm{d} {\\Phi} -a {\\mu}^{2} \\mathrm{d} {\\Psi}$$" ], "text/plain": [ "dt + (a*mu^2 - a) dPh - a*mu^2 dPs" ] }, "execution_count": 37, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = dt - a*sinth2*dPh - a*costh2*dPs\n", "s.display()" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( -\\frac{1}{\\sqrt{{\\mu} + 1} \\sqrt{-{\\mu} + 1}} \\right) \\mathrm{d} {\\mu}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( -\\frac{1}{\\sqrt{{\\mu} + 1} \\sqrt{-{\\mu} + 1}} \\right) \\mathrm{d} {\\mu}$$" ], "text/plain": [ "-1/(sqrt(mu + 1)*sqrt(-mu + 1)) dmu" ] }, "execution_count": 38, "metadata": { }, "output_type": "execute_result" } ], "source": [ "dth.display()" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 39, "metadata": { }, "output_type": "execute_result" } ], "source": [ "G1 = - (1 + y^2*l^2)* dt*dt \\\n", " + y^2*(dth*dth + sinth2* dPh*dPh + costh2* dPs*dPs) \\\n", " + 2*m/(y^2*Xi_a^3)* s*s \\\n", " + y^4/(y^4*(1 + y^2*l^2) - 2*m*y^2/Xi_a^2 + 2*m*a^2/Xi_a^3)* dy*dy\n", "# NB: note the Xi_a^3 term in the factor of s*s differs from Eq. (5.32) of Hawking et al (1999)\n", "G == G1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "## String worldsheet" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "The string worldsheet as a 2-dimensional pseudo-Riemannian manifold (we don't assume Lorentzian signature here):" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2-dimensional Riemannian manifold W\n" ] } ], "source": [ "W = Manifold(2, 'W', structure='pseudo-Riemannian')\n", "print(W)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "Let us assume that the string worldsheet is parametrized by $(t,y)$:" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(W,(t, y)\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(W,(t, y)\\right)$$" ], "text/plain": [ "Chart (W, (t, y))" ] }, "execution_count": 41, "metadata": { }, "output_type": "execute_result" } ], "source": [ "XW. = W.chart(r't y:(a/sqrt(1-a^2*l^2),+oo)')\n", "XW" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "The string embedding in Kerr-AdS spacetime, as an expansion about a straight string solution in AdS (Eqs. (4.30)-(4.32) of the paper)" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} F:& W & \\longrightarrow & \\mathcal{M} \\\\ & \\left(t, y\\right) & \\longmapsto & \\left(t, r, {\\mu}, {\\phi}, {\\psi}\\right) = \\left(t, \\sqrt{-{\\left(a^{2} {\\ell}^{2} - 1\\right)} y^{2} - a^{2}}, a^{2} \\mu_{1}\\left(y\\right) + {\\mu_0}, {\\left(a {\\beta_1} - a\\right)} {\\ell}^{2} t + a {\\beta_1} \\Phi_{1}\\left(y\\right) + {\\Phi_0}, {\\left(a {\\beta_2} - a\\right)} {\\ell}^{2} t + a {\\beta_2} \\Psi_{1}\\left(y\\right) + {\\Psi_0}\\right) \\\\ & \\left(t, y\\right) & \\longmapsto & \\left(t, y, {\\mu}, {\\Phi}, {\\Psi}\\right) = \\left(t, y, a^{2} \\mu_{1}\\left(y\\right) + {\\mu_0}, a {\\beta_1} {\\ell}^{2} t + a {\\beta_1} \\Phi_{1}\\left(y\\right) + {\\Phi_0}, a {\\beta_2} {\\ell}^{2} t + a {\\beta_2} \\Psi_{1}\\left(y\\right) + {\\Psi_0}\\right) \\end{array}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} F:& W & \\longrightarrow & \\mathcal{M} \\\\ & \\left(t, y\\right) & \\longmapsto & \\left(t, r, {\\mu}, {\\phi}, {\\psi}\\right) = \\left(t, \\sqrt{-{\\left(a^{2} {\\ell}^{2} - 1\\right)} y^{2} - a^{2}}, a^{2} \\mu_{1}\\left(y\\right) + {\\mu_0}, {\\left(a {\\beta_1} - a\\right)} {\\ell}^{2} t + a {\\beta_1} \\Phi_{1}\\left(y\\right) + {\\Phi_0}, {\\left(a {\\beta_2} - a\\right)} {\\ell}^{2} t + a {\\beta_2} \\Psi_{1}\\left(y\\right) + {\\Psi_0}\\right) \\\\ & \\left(t, y\\right) & \\longmapsto & \\left(t, y, {\\mu}, {\\Phi}, {\\Psi}\\right) = \\left(t, y, a^{2} \\mu_{1}\\left(y\\right) + {\\mu_0}, a {\\beta_1} {\\ell}^{2} t + a {\\beta_1} \\Phi_{1}\\left(y\\right) + {\\Phi_0}, a {\\beta_2} {\\ell}^{2} t + a {\\beta_2} \\Psi_{1}\\left(y\\right) + {\\Psi_0}\\right) \\end{array}$$" ], "text/plain": [ "F: W --> M\n", " (t, y) |--> (t, r, mu, ph, ps) = (t, sqrt(-(a^2*l^2 - 1)*y^2 - a^2), a^2*mu_1(y) + Mu0, (a*beta1 - a)*l^2*t + a*beta1*Phi_1(y) + Phi0, (a*beta2 - a)*l^2*t + a*beta2*Psi_1(y) + Psi0)\n", " (t, y) |--> (t, y, mu, Ph, Ps) = (t, y, a^2*mu_1(y) + Mu0, a*beta1*l^2*t + a*beta1*Phi_1(y) + Phi0, a*beta2*l^2*t + a*beta2*Psi_1(y) + Psi0)" ] }, "execution_count": 42, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Mu0 = var('Mu0', latex_name=r'\\mu_0', domain='real')\n", "Phi0 = var('Phi0', latex_name=r'\\Phi_0', domain='real')\n", "Psi0 = var('Psi0', latex_name=r'\\Psi_0', domain='real')\n", "beta1 = var('beta1', latex_name=r'\\beta_1', domain='real')\n", "beta2 = var('beta2', latex_name=r'\\beta_2', domain='real')\n", "\n", "cosTh0 = Mu0\n", "sinTh0 = sqrt(1 - Mu0^2)\n", "\n", "mu_s = Mu0 + a^2*function('mu_1')(y)\n", "Ph_s = Phi0 + beta1*a*l^2*t + beta1*a*function('Phi_1')(y)\n", "Ps_s = Psi0 + beta2*a*l^2*t + beta2*a*function('Psi_1')(y)\n", "\n", "F = W.diff_map(M, {(XW, ADS): [t, y, mu_s, Ph_s, Ps_s]}, name='F') \n", "F.display()" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n", "1 & 0 \\\\\n", "0 & 1 \\\\\n", "0 & a^{2} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) \\\\\n", "a {\\beta_1} {\\ell}^{2} & a {\\beta_1} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) \\\\\n", "a {\\beta_2} {\\ell}^{2} & a {\\beta_2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)\n", "\\end{array}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n", "1 & 0 \\\\\n", "0 & 1 \\\\\n", "0 & a^{2} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) \\\\\n", "a {\\beta_1} {\\ell}^{2} & a {\\beta_1} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) \\\\\n", "a {\\beta_2} {\\ell}^{2} & a {\\beta_2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)\n", "\\end{array}\\right)$$" ], "text/plain": [ "[ 1 0]\n", "[ 0 1]\n", "[ 0 a^2*diff(mu_1(y), y)]\n", "[ a*beta1*l^2 a*beta1*diff(Phi_1(y), y)]\n", "[ a*beta2*l^2 a*beta2*diff(Psi_1(y), y)]" ] }, "execution_count": 43, "metadata": { }, "output_type": "execute_result" } ], "source": [ "F.jacobian_matrix()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "size": 4 }, "source": [ "### Induced metric on the string worldsheet\n", "\n", "The string worldsheet metric is the metric $g$ induced by the spacetime metric $G$, i.e. the pullback of $G$ by the embedding $F$:" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ ], "source": [ "g = W.metric()\n", "g.set(F.pullback(G))" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": false, "size": 4 }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {\\left(a^{12} {\\beta_1}^{2} - 2 \\, a^{12} {\\beta_1} {\\beta_2} + a^{12} {\\beta_2}^{2}\\right)} {\\ell}^{4} m \\mu_{1}\\left(y\\right)^{4} + 8 \\, {\\left({\\mu_0} a^{10} {\\beta_1}^{2} - 2 \\, {\\mu_0} a^{10} {\\beta_1} {\\beta_2} + {\\mu_0} a^{10} {\\beta_2}^{2}\\right)} {\\ell}^{4} m \\mu_{1}\\left(y\\right)^{3} - {\\left({\\left({\\mu_0}^{2} a^{8} {\\beta_2}^{2} - {\\left({\\mu_0}^{2} - 1\\right)} a^{8} {\\beta_1}^{2}\\right)} {\\ell}^{10} - {\\left(3 \\, {\\mu_0}^{2} a^{6} {\\beta_2}^{2} - 3 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{6} {\\beta_1}^{2} + a^{6}\\right)} {\\ell}^{8} + 3 \\, {\\left({\\mu_0}^{2} a^{4} {\\beta_2}^{2} - {\\left({\\mu_0}^{2} - 1\\right)} a^{4} {\\beta_1}^{2} + a^{4}\\right)} {\\ell}^{6} - {\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} + 3 \\, a^{2}\\right)} {\\ell}^{4} + {\\ell}^{2}\\right)} y^{4} + {\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2} + {\\left({\\left({\\left(a^{12} {\\beta_1}^{2} - a^{12} {\\beta_2}^{2}\\right)} {\\ell}^{10} - 3 \\, {\\left(a^{10} {\\beta_1}^{2} - a^{10} {\\beta_2}^{2}\\right)} {\\ell}^{8} + 3 \\, {\\left(a^{8} {\\beta_1}^{2} - a^{8} {\\beta_2}^{2}\\right)} {\\ell}^{6} - {\\left(a^{6} {\\beta_1}^{2} - a^{6} {\\beta_2}^{2}\\right)} {\\ell}^{4}\\right)} y^{4} + 4 \\, {\\left({\\left(3 \\, {\\mu_0}^{2} a^{8} {\\beta_2}^{2} + {\\left(3 \\, {\\mu_0}^{2} - 1\\right)} a^{8} {\\beta_1}^{2} - {\\left(6 \\, {\\mu_0}^{2} - 1\\right)} a^{8} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{4} + {\\left(a^{6} {\\beta_1} - a^{6} {\\beta_2}\\right)} {\\ell}^{2}\\right)} m\\right)} \\mu_{1}\\left(y\\right)^{2} + 2 \\, {\\left({\\left({\\mu_0}^{4} a^{4} {\\beta_2}^{2} + {\\left({\\mu_0}^{4} - 2 \\, {\\mu_0}^{2} + 1\\right)} a^{4} {\\beta_1}^{2} - 2 \\, {\\left({\\mu_0}^{4} - {\\mu_0}^{2}\\right)} a^{4} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{4} - 2 \\, {\\left({\\mu_0}^{2} a^{2} {\\beta_2} - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}\\right)} {\\ell}^{2} + 1\\right)} m + 2 \\, {\\left({\\left({\\left({\\mu_0} a^{10} {\\beta_1}^{2} - {\\mu_0} a^{10} {\\beta_2}^{2}\\right)} {\\ell}^{10} - 3 \\, {\\left({\\mu_0} a^{8} {\\beta_1}^{2} - {\\mu_0} a^{8} {\\beta_2}^{2}\\right)} {\\ell}^{8} + 3 \\, {\\left({\\mu_0} a^{6} {\\beta_1}^{2} - {\\mu_0} a^{6} {\\beta_2}^{2}\\right)} {\\ell}^{6} - {\\left({\\mu_0} a^{4} {\\beta_1}^{2} - {\\mu_0} a^{4} {\\beta_2}^{2}\\right)} {\\ell}^{4}\\right)} y^{4} + 4 \\, {\\left({\\left({\\mu_0}^{3} a^{6} {\\beta_2}^{2} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{6} {\\beta_1}^{2} - {\\left(2 \\, {\\mu_0}^{3} - {\\mu_0}\\right)} a^{6} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{4} + {\\left({\\mu_0} a^{4} {\\beta_1} - {\\mu_0} a^{4} {\\beta_2}\\right)} {\\ell}^{2}\\right)} m\\right)} \\mu_{1}\\left(y\\right)}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, {\\left(a^{12} {\\beta_1}^{2} - 2 \\, a^{12} {\\beta_1} {\\beta_2} + a^{12} {\\beta_2}^{2}\\right)} {\\ell}^{4} m \\mu_{1}\\left(y\\right)^{4} + 8 \\, {\\left({\\mu_0} a^{10} {\\beta_1}^{2} - 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3 \\, {\\mu_0}^{2} a^{6} {\\beta_2}^{2} {\\ell}^{6} + 3 \\, {\\mu_0}^{2} a^{4} {\\beta_2}^{2} {\\ell}^{4} - {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2}\\right)} y^{4} + {\\left({\\left(a^{12} {\\beta_2}^{2} {\\ell}^{8} - 3 \\, a^{10} {\\beta_2}^{2} {\\ell}^{6} + 3 \\, a^{8} {\\beta_2}^{2} {\\ell}^{4} - a^{6} {\\beta_2}^{2} {\\ell}^{2}\\right)} y^{4} + 2 \\, {\\left(a^{6} {\\beta_2} - {\\left(6 \\, {\\mu_0}^{2} a^{8} {\\beta_2}^{2} - {\\left(6 \\, {\\mu_0}^{2} - 1\\right)} a^{8} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{2}\\right)} m\\right)} \\mu_{1}\\left(y\\right)^{2} + 2 \\, {\\left({\\mu_0}^{2} a^{2} {\\beta_2} - {\\left({\\mu_0}^{4} a^{4} {\\beta_2}^{2} - {\\left({\\mu_0}^{4} - {\\mu_0}^{2}\\right)} a^{4} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{2}\\right)} m + 2 \\, {\\left({\\left({\\mu_0} a^{10} {\\beta_2}^{2} {\\ell}^{8} - 3 \\, {\\mu_0} a^{8} {\\beta_2}^{2} {\\ell}^{6} + 3 \\, {\\mu_0} a^{6} {\\beta_2}^{2} {\\ell}^{4} - {\\mu_0} a^{4} {\\beta_2}^{2} {\\ell}^{2}\\right)} y^{4} + 2 \\, {\\left({\\mu_0} a^{4} {\\beta_2} - {\\left(2 \\, {\\mu_0}^{3} a^{6} {\\beta_2}^{2} - {\\left(2 \\, {\\mu_0}^{3} - {\\mu_0}\\right)} a^{6} {\\beta_1} {\\beta_2}\\right)} {\\ell}^{2}\\right)} m\\right)} \\mu_{1}\\left(y\\right)\\right)} \\frac{\\partial\\,\\Psi_{1}}{\\partial y}}{{\\left(a^{6} {\\ell}^{6} - 3 \\, a^{4} {\\ell}^{4} + 3 \\, a^{2} {\\ell}^{2} - 1\\right)} y^{2}}$$" ], "text/plain": [ "-((2*(a^12*beta1^2 - a^12*beta1*beta2)*l^2*m*mu_1(y)^4 + 8*(Mu0*a^10*beta1^2 - Mu0*a^10*beta1*beta2)*l^2*m*mu_1(y)^3 + ((Mu0^2 - 1)*a^8*beta1^2*l^8 - 3*(Mu0^2 - 1)*a^6*beta1^2*l^6 + 3*(Mu0^2 - 1)*a^4*beta1^2*l^4 - (Mu0^2 - 1)*a^2*beta1^2*l^2)*y^4 + ((a^12*beta1^2*l^8 - 3*a^10*beta1^2*l^6 + 3*a^8*beta1^2*l^4 - a^6*beta1^2*l^2)*y^4 + 2*(a^6*beta1 + (2*(3*Mu0^2 - 1)*a^8*beta1^2 - (6*Mu0^2 - 1)*a^8*beta1*beta2)*l^2)*m)*mu_1(y)^2 + 2*((Mu0^2 - 1)*a^2*beta1 + ((Mu0^4 - 2*Mu0^2 + 1)*a^4*beta1^2 - (Mu0^4 - Mu0^2)*a^4*beta1*beta2)*l^2)*m + 2*((Mu0*a^10*beta1^2*l^8 - 3*Mu0*a^8*beta1^2*l^6 + 3*Mu0*a^6*beta1^2*l^4 - Mu0*a^4*beta1^2*l^2)*y^4 + 2*(Mu0*a^4*beta1 + (2*(Mu0^3 - Mu0)*a^6*beta1^2 - (2*Mu0^3 - Mu0)*a^6*beta1*beta2)*l^2)*m)*mu_1(y))*d(Phi_1)/dy - (2*(a^12*beta1*beta2 - a^12*beta2^2)*l^2*m*mu_1(y)^4 + 8*(Mu0*a^10*beta1*beta2 - Mu0*a^10*beta2^2)*l^2*m*mu_1(y)^3 + (Mu0^2*a^8*beta2^2*l^8 - 3*Mu0^2*a^6*beta2^2*l^6 + 3*Mu0^2*a^4*beta2^2*l^4 - Mu0^2*a^2*beta2^2*l^2)*y^4 + ((a^12*beta2^2*l^8 - 3*a^10*beta2^2*l^6 + 3*a^8*beta2^2*l^4 - a^6*beta2^2*l^2)*y^4 + 2*(a^6*beta2 - (6*Mu0^2*a^8*beta2^2 - (6*Mu0^2 - 1)*a^8*beta1*beta2)*l^2)*m)*mu_1(y)^2 + 2*(Mu0^2*a^2*beta2 - (Mu0^4*a^4*beta2^2 - (Mu0^4 - Mu0^2)*a^4*beta1*beta2)*l^2)*m + 2*((Mu0*a^10*beta2^2*l^8 - 3*Mu0*a^8*beta2^2*l^6 + 3*Mu0*a^6*beta2^2*l^4 - Mu0*a^4*beta2^2*l^2)*y^4 + 2*(Mu0*a^4*beta2 - (2*Mu0^3*a^6*beta2^2 - (2*Mu0^3 - Mu0)*a^6*beta1*beta2)*l^2)*m)*mu_1(y))*d(Psi_1)/dy)/((a^6*l^6 - 3*a^4*l^4 + 3*a^2*l^2 - 1)*y^2)" ] }, "execution_count": 47, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g[0,1]" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Nambu-Goto action" ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "detg = g.determinant().expr()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Expanding at second order in $a$:" ] }, { "cell_type": "code", "execution_count": 49, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2}\\right)} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{6} - 2 \\, {\\left({\\left(2 \\, {\\mu_0}^{2} a^{2} {\\beta_2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1} - a^{2}\\right)} {\\ell}^{2} - 1\\right)} m y^{2} - y^{4} + 2 \\, a^{2} m + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{10} + 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{8} - 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m y^{4} + 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m^{2} y^{2} - {\\left(4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2}\\right)} y^{6}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - {\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{4} y^{10} + 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{8} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m y^{4} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m^{2} y^{2} - {\\left(4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} m - {\\mu_0}^{2} a^{2} {\\beta_2}^{2}\\right)} y^{6}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2}}{{\\ell}^{2} y^{6} + y^{4} - 2 \\, m y^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2}\\right)} {\\ell}^{4} - {\\ell}^{2}\\right)} y^{6} - 2 \\, {\\left({\\left(2 \\, {\\mu_0}^{2} a^{2} {\\beta_2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1} - a^{2}\\right)} {\\ell}^{2} - 1\\right)} m y^{2} - y^{4} + 2 \\, a^{2} m + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{10} + 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{8} - 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m y^{4} + 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m^{2} y^{2} - {\\left(4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2}\\right)} y^{6}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - {\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{4} y^{10} + 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{8} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m y^{4} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m^{2} y^{2} - {\\left(4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} m - {\\mu_0}^{2} a^{2} {\\beta_2}^{2}\\right)} y^{6}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2}}{{\\ell}^{2} y^{6} + y^{4} - 2 \\, m y^{2}}$$" ], "text/plain": [ "(((Mu0^2*a^2*beta2^2 - (Mu0^2 - 1)*a^2*beta1^2)*l^4 - l^2)*y^6 - 2*((2*Mu0^2*a^2*beta2 - 2*(Mu0^2 - 1)*a^2*beta1 - a^2)*l^2 - 1)*m*y^2 - y^4 + 2*a^2*m + ((Mu0^2 - 1)*a^2*beta1^2*l^4*y^10 + 2*(Mu0^2 - 1)*a^2*beta1^2*l^2*y^8 - 4*(Mu0^2 - 1)*a^2*beta1^2*m*y^4 + 4*(Mu0^2 - 1)*a^2*beta1^2*m^2*y^2 - (4*(Mu0^2 - 1)*a^2*beta1^2*l^2*m - (Mu0^2 - 1)*a^2*beta1^2)*y^6)*diff(Phi_1(y), y)^2 - (Mu0^2*a^2*beta2^2*l^4*y^10 + 2*Mu0^2*a^2*beta2^2*l^2*y^8 - 4*Mu0^2*a^2*beta2^2*m*y^4 + 4*Mu0^2*a^2*beta2^2*m^2*y^2 - (4*Mu0^2*a^2*beta2^2*l^2*m - Mu0^2*a^2*beta2^2)*y^6)*diff(Psi_1(y), y)^2)/(l^2*y^6 + y^4 - 2*m*y^2)" ] }, "execution_count": 49, "metadata": { }, "output_type": "execute_result" } ], "source": [ "detg_a2 = detg.series(a, 3).truncate().simplify_full()\n", "detg_a2 " ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The Nambu-Goto Lagrangian at second order in $a$:" ] }, { "cell_type": "code", "execution_count": 50, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left({\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2}\\right)} {\\ell}^{4} - 2 \\, {\\ell}^{2}\\right)} y^{6} - 2 \\, {\\left({\\left(2 \\, {\\mu_0}^{2} a^{2} {\\beta_2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1} - a^{2}\\right)} {\\ell}^{2} - 2\\right)} m y^{2} - 2 \\, y^{4} + 2 \\, a^{2} m + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{10} + 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{8} - 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m y^{4} + 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m^{2} y^{2} - {\\left(4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2}\\right)} y^{6}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - {\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{4} y^{10} + 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{8} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m y^{4} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m^{2} y^{2} - {\\left(4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} m - {\\mu_0}^{2} a^{2} {\\beta_2}^{2}\\right)} y^{6}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2}}{2 \\, {\\left({\\ell}^{2} y^{6} + y^{4} - 2 \\, m y^{2}\\right)}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left({\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2}\\right)} {\\ell}^{4} - 2 \\, {\\ell}^{2}\\right)} y^{6} - 2 \\, {\\left({\\left(2 \\, {\\mu_0}^{2} a^{2} {\\beta_2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1} - a^{2}\\right)} {\\ell}^{2} - 2\\right)} m y^{2} - 2 \\, y^{4} + 2 \\, a^{2} m + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{10} + 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{8} - 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m y^{4} + 4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m^{2} y^{2} - {\\left(4 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2}\\right)} y^{6}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - {\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{4} y^{10} + 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{8} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m y^{4} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m^{2} y^{2} - {\\left(4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} m - {\\mu_0}^{2} a^{2} {\\beta_2}^{2}\\right)} y^{6}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2}}{2 \\, {\\left({\\ell}^{2} y^{6} + y^{4} - 2 \\, m y^{2}\\right)}}$$" ], "text/plain": [ "-1/2*(((Mu0^2*a^2*beta2^2 - (Mu0^2 - 1)*a^2*beta1^2)*l^4 - 2*l^2)*y^6 - 2*((2*Mu0^2*a^2*beta2 - 2*(Mu0^2 - 1)*a^2*beta1 - a^2)*l^2 - 2)*m*y^2 - 2*y^4 + 2*a^2*m + ((Mu0^2 - 1)*a^2*beta1^2*l^4*y^10 + 2*(Mu0^2 - 1)*a^2*beta1^2*l^2*y^8 - 4*(Mu0^2 - 1)*a^2*beta1^2*m*y^4 + 4*(Mu0^2 - 1)*a^2*beta1^2*m^2*y^2 - (4*(Mu0^2 - 1)*a^2*beta1^2*l^2*m - (Mu0^2 - 1)*a^2*beta1^2)*y^6)*diff(Phi_1(y), y)^2 - (Mu0^2*a^2*beta2^2*l^4*y^10 + 2*Mu0^2*a^2*beta2^2*l^2*y^8 - 4*Mu0^2*a^2*beta2^2*m*y^4 + 4*Mu0^2*a^2*beta2^2*m^2*y^2 - (4*Mu0^2*a^2*beta2^2*l^2*m - Mu0^2*a^2*beta2^2)*y^6)*diff(Psi_1(y), y)^2)/(l^2*y^6 + y^4 - 2*m*y^2)" ] }, "execution_count": 50, "metadata": { }, "output_type": "execute_result" } ], "source": [ "L_a2 = (sqrt(-detg_a2)).series(a, 3).truncate().simplify_full()\n", "L_a2" ] }, { "cell_type": "code", "execution_count": 51, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\mu_0}^{2} a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{10} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{4} y^{10} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{10} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 2 \\, {\\mu_0}^{2} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{8} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{8} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_1}^{2} {\\ell}^{2} m y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} m y^{6} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + {\\mu_0}^{2} a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{6} - {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{4} y^{6} + 2 \\, a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{8} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, a^{2} {\\beta_1}^{2} {\\ell}^{2} m y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{6} - {\\mu_0}^{2} a^{2} {\\beta_1}^{2} y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y^{6} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_1}^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_1}^{2} m^{2} y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + a^{2} {\\beta_1}^{2} y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m^{2} y^{2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 4 \\, a^{2} {\\beta_1}^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_1} {\\ell}^{2} m y^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2} {\\ell}^{2} m y^{2} + 4 \\, a^{2} {\\beta_1}^{2} m^{2} y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 4 \\, a^{2} {\\beta_1} {\\ell}^{2} m y^{2} + 2 \\, {\\ell}^{2} y^{6} - 2 \\, a^{2} {\\ell}^{2} m y^{2} + 2 \\, y^{4} - 2 \\, a^{2} m - 4 \\, m y^{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\mu_0}^{2} a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{10} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{4} y^{10} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{10} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 2 \\, {\\mu_0}^{2} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{8} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{8} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_1}^{2} {\\ell}^{2} m y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} m y^{6} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + {\\mu_0}^{2} a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{6} - {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{4} y^{6} + 2 \\, a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{8} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, a^{2} {\\beta_1}^{2} {\\ell}^{2} m y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - a^{2} {\\beta_1}^{2} {\\ell}^{4} y^{6} - {\\mu_0}^{2} a^{2} {\\beta_1}^{2} y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y^{6} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_1}^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_1}^{2} m^{2} y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + a^{2} {\\beta_1}^{2} y^{6} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m^{2} y^{2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} - 4 \\, a^{2} {\\beta_1}^{2} m y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} - 4 \\, {\\mu_0}^{2} a^{2} {\\beta_1} {\\ell}^{2} m y^{2} + 4 \\, {\\mu_0}^{2} a^{2} {\\beta_2} {\\ell}^{2} m y^{2} + 4 \\, a^{2} {\\beta_1}^{2} m^{2} y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + 4 \\, a^{2} {\\beta_1} {\\ell}^{2} m y^{2} + 2 \\, {\\ell}^{2} y^{6} - 2 \\, a^{2} {\\ell}^{2} m y^{2} + 2 \\, y^{4} - 2 \\, a^{2} m - 4 \\, m y^{2}$$" ], "text/plain": [ "-Mu0^2*a^2*beta1^2*l^4*y^10*diff(Phi_1(y), y)^2 + Mu0^2*a^2*beta2^2*l^4*y^10*diff(Psi_1(y), y)^2 + a^2*beta1^2*l^4*y^10*diff(Phi_1(y), y)^2 - 2*Mu0^2*a^2*beta1^2*l^2*y^8*diff(Phi_1(y), y)^2 + 2*Mu0^2*a^2*beta2^2*l^2*y^8*diff(Psi_1(y), y)^2 + 4*Mu0^2*a^2*beta1^2*l^2*m*y^6*diff(Phi_1(y), y)^2 - 4*Mu0^2*a^2*beta2^2*l^2*m*y^6*diff(Psi_1(y), y)^2 + Mu0^2*a^2*beta1^2*l^4*y^6 - Mu0^2*a^2*beta2^2*l^4*y^6 + 2*a^2*beta1^2*l^2*y^8*diff(Phi_1(y), y)^2 - 4*a^2*beta1^2*l^2*m*y^6*diff(Phi_1(y), y)^2 - a^2*beta1^2*l^4*y^6 - Mu0^2*a^2*beta1^2*y^6*diff(Phi_1(y), y)^2 + Mu0^2*a^2*beta2^2*y^6*diff(Psi_1(y), y)^2 + 4*Mu0^2*a^2*beta1^2*m*y^4*diff(Phi_1(y), y)^2 - 4*Mu0^2*a^2*beta2^2*m*y^4*diff(Psi_1(y), y)^2 - 4*Mu0^2*a^2*beta1^2*m^2*y^2*diff(Phi_1(y), y)^2 + a^2*beta1^2*y^6*diff(Phi_1(y), y)^2 + 4*Mu0^2*a^2*beta2^2*m^2*y^2*diff(Psi_1(y), y)^2 - 4*a^2*beta1^2*m*y^4*diff(Phi_1(y), y)^2 - 4*Mu0^2*a^2*beta1*l^2*m*y^2 + 4*Mu0^2*a^2*beta2*l^2*m*y^2 + 4*a^2*beta1^2*m^2*y^2*diff(Phi_1(y), y)^2 + 4*a^2*beta1*l^2*m*y^2 + 2*l^2*y^6 - 2*a^2*l^2*m*y^2 + 2*y^4 - 2*a^2*m - 4*m*y^2" ] }, "execution_count": 51, "metadata": { }, "output_type": "execute_result" } ], "source": [ "L_a2.numerator()" ] }, { "cell_type": "code", "execution_count": 52, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\ell}^{2} y^{6} + 2 \\, y^{4} - 4 \\, m y^{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\ell}^{2} y^{6} + 2 \\, y^{4} - 4 \\, m y^{2}$$" ], "text/plain": [ "2*l^2*y^6 + 2*y^4 - 4*m*y^2" ] }, "execution_count": 52, "metadata": { }, "output_type": "execute_result" } ], "source": [ "L_a2.denominator()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Euler-Lagrange equations" ] }, { "cell_type": "code", "execution_count": 53, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def euler_lagrange(lagr, qs, var):\n", " r\"\"\"\n", " Derive the Euler-Lagrange equations from a given Lagrangian.\n", "\n", " INPUT:\n", "\n", " - ``lagr`` -- symbolic expression representing the Lagrangian density\n", " - ``qs`` -- either a single symbolic function or a list/tuple of\n", " symbolic functions, representing the `q`'s; these functions must\n", " appear in ``lagr`` up to at most their first derivatives\n", " - ``var`` -- either a single variable, typically `t` (1-dimensional\n", " problem) or a list/tuple of symbolic variables\n", "\n", " OUTPUT:\n", "\n", " - list of Euler-Lagrange equations; if only one function is involved, the\n", " single Euler-Lagrannge equation is returned instead.\n", "\n", " \"\"\"\n", " if not isinstance(qs, (list, tuple)):\n", " qs = [qs]\n", " if not isinstance(var, (list, tuple)):\n", " var = [var]\n", " n = len(qs)\n", " d = len(var)\n", " qv = [SR.var('qxxxx{}'.format(q)) for q in qs]\n", " dqv = [[SR.var('qxxxx{}_{}'.format(q, v)) for v in var] for q in qs]\n", " subs = {qs[i](*var): qv[i] for i in range(n)}\n", " subs_inv = {qv[i]: qs[i](*var) for i in range(n)}\n", " for i in range(n):\n", " subs.update({diff(qs[i](*var), var[j]): dqv[i][j]\n", " for j in range(d)})\n", " subs_inv.update({dqv[i][j]: diff(qs[i](*var), var[j])\n", " for j in range(d)})\n", " lg = lagr.substitute(subs)\n", " eqs = []\n", " for i in range(n):\n", " dLdq = diff(lg, qv[i]).simplify_full()\n", " dLdq = dLdq.substitute(subs_inv)\n", " ddL = 0\n", " for j in range(d):\n", " h = diff(lg, dqv[i][j]).simplify_full()\n", " h = h.substitute(subs_inv)\n", " ddL += diff(h, var[j])\n", " eqs.append((dLdq - ddL).simplify_full() == 0)\n", " if n == 1:\n", " return eqs[0]\n", " return eqs" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We compute the Euler-Lagrange equations at order $a^2$ for $\\phi_1$ and $\\psi_1$:" ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, {\\left(2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{3} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{4} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0, -2 \\, {\\left(2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{3} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{4} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y^{2} - 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0\\right]$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, {\\left(2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{3} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{4} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0, -2 \\, {\\left(2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{3} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{4} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y^{2} - 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0\\right]$$" ], "text/plain": [ "[2*(2*(Mu0^2 - 1)*a^2*beta1^2*l^2*y^3 + (Mu0^2 - 1)*a^2*beta1^2*y)*diff(Phi_1(y), y) + ((Mu0^2 - 1)*a^2*beta1^2*l^2*y^4 + (Mu0^2 - 1)*a^2*beta1^2*y^2 - 2*(Mu0^2 - 1)*a^2*beta1^2*m)*diff(Phi_1(y), y, y) == 0,\n", " -2*(2*Mu0^2*a^2*beta2^2*l^2*y^3 + Mu0^2*a^2*beta2^2*y)*diff(Psi_1(y), y) - (Mu0^2*a^2*beta2^2*l^2*y^4 + Mu0^2*a^2*beta2^2*y^2 - 2*Mu0^2*a^2*beta2^2*m)*diff(Psi_1(y), y, y) == 0]" ] }, "execution_count": 54, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eqs = euler_lagrange(L_a2, [Phi_1, Psi_1], y)\n", "eqs" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "#### Solving the equation for $\\phi_1$ (check of Eq. (4.34))" ] }, { "cell_type": "code", "execution_count": 55, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{3} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{4} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{3} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left({\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{4} + {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0$$" ], "text/plain": [ "2*(2*(Mu0^2 - 1)*a^2*beta1^2*l^2*y^3 + (Mu0^2 - 1)*a^2*beta1^2*y)*diff(Phi_1(y), y) + ((Mu0^2 - 1)*a^2*beta1^2*l^2*y^4 + (Mu0^2 - 1)*a^2*beta1^2*y^2 - 2*(Mu0^2 - 1)*a^2*beta1^2*m)*diff(Phi_1(y), y, y) == 0" ] }, "execution_count": 55, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_phi1 = eqs[0]\n", "eq_phi1" ] }, { "cell_type": "code", "execution_count": 56, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\ell}^{2} y^{3} + y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, {\\left(2 \\, {\\ell}^{2} y^{3} + y\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + {\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Phi_{1}\\left(y\\right) = 0$$" ], "text/plain": [ "2*(2*l^2*y^3 + y)*diff(Phi_1(y), y) + (l^2*y^4 + y^2 - 2*m)*diff(Phi_1(y), y, y) == 0" ] }, "execution_count": 56, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_phi1 = (eq_phi1/(a^2*(Mu0^2-1)*beta1^2)).simplify_full()\n", "eq_phi1" ] }, { "cell_type": "code", "execution_count": 57, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{1} \\int \\frac{1}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} + K_{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{1} \\int \\frac{1}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} + K_{2}$$" ], "text/plain": [ "_K1*integrate(1/(l^2*y^4 + y^2 - 2*m), y) + _K2" ] }, "execution_count": 57, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Phi1_sol(y) = desolve(eq_phi1, Phi_1(y), ivar=y)\n", "Phi1_sol(y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The symbolic constants $K_1$ and $K_2$ are actually denoted `_K1` and `_K2` by SageMath, as the `print` reveals:" ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "_K1*integrate(1/(l^2*y^4 + y^2 - 2*m), y) + _K2\n" ] } ], "source": [ "print(Phi1_sol(y))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Hence we perform the substitutions with `SR.var('_K1')` and `SR.var('_K2')`:" ] }, { "cell_type": "code", "execution_count": 59, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "P*integrate(1/(l^2*y^4 + y^2 - 2*m), y)\n" ] } ], "source": [ "P = var(\"P\", latex_name=r\"\\mathcal{P}'\")\n", "Phi1_sol(y) = Phi1_sol(y).subs({SR.var('_K1'): P, SR.var('_K2'): 0})\n", "print(Phi1_sol(y))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "#### Solving the equation for $\\psi_1$ (check of Eq. (4.34))" ] }, { "cell_type": "code", "execution_count": 60, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{3} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{4} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y^{2} - 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{3} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left({\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{4} + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y^{2} - 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0$$" ], "text/plain": [ "-2*(2*Mu0^2*a^2*beta2^2*l^2*y^3 + Mu0^2*a^2*beta2^2*y)*diff(Psi_1(y), y) - (Mu0^2*a^2*beta2^2*l^2*y^4 + Mu0^2*a^2*beta2^2*y^2 - 2*Mu0^2*a^2*beta2^2*m)*diff(Psi_1(y), y, y) == 0" ] }, "execution_count": 60, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_psi1 = eqs[1]\n", "eq_psi1" ] }, { "cell_type": "code", "execution_count": 61, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(2 \\, {\\ell}^{2} y^{3} + y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, {\\left(2 \\, {\\ell}^{2} y^{3} + y\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - {\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\Psi_{1}\\left(y\\right) = 0$$" ], "text/plain": [ "-2*(2*l^2*y^3 + y)*diff(Psi_1(y), y) - (l^2*y^4 + y^2 - 2*m)*diff(Psi_1(y), y, y) == 0" ] }, "execution_count": 61, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_psi1 = (eq_psi1/(a^2*Mu0^2*beta2^2)).simplify_full()\n", "eq_psi1" ] }, { "cell_type": "code", "execution_count": 62, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{1} \\int \\frac{1}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} + K_{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{1} \\int \\frac{1}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} + K_{2}$$" ], "text/plain": [ "_K1*integrate(1/(l^2*y^4 + y^2 - 2*m), y) + _K2" ] }, "execution_count": 62, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Psi1_sol(y) = desolve(eq_psi1, Psi_1(y), ivar=y)\n", "Psi1_sol(y)" ] }, { "cell_type": "code", "execution_count": 63, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Q*integrate(1/(l^2*y^4 + y^2 - 2*m), y)\n" ] } ], "source": [ "Q = var('Q', latex_name=r\"\\mathcal{Q}'\")\n", "Psi1_sol(y) = Psi1_sol(y).subs({SR.var('_K1'): Q, SR.var('_K2'): 0})\n", "print(Psi1_sol(y))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Nambu-Goto Lagrangian at fourth order in $a$" ] }, { "cell_type": "code", "execution_count": 64, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "detg_a4 = detg.series(a, 5).truncate().simplify_full()" ] }, { "cell_type": "code", "execution_count": 65, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "L_a4 = (sqrt(-detg_a4)).series(a, 5).truncate().simplify_full()" ] }, { "cell_type": "code", "execution_count": 66, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "eqs = euler_lagrange(L_a4, [Phi_1, Psi_1, mu_1], y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### The equation for $\\mu_1$ (check of Eq. (4.35))" ] }, { "cell_type": "code", "execution_count": 67, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}\\right)} {\\ell}^{2} m - {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} {\\ell}^{4} y^{8} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} {\\ell}^{2} y^{6} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} m y^{2} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} m^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} {\\ell}^{4} y^{8} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} {\\ell}^{2} y^{6} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} m y^{2} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} m^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 2 \\, {\\left(2 \\, a^{4} {\\ell}^{4} y^{7} + 3 \\, a^{4} {\\ell}^{2} y^{5} - 2 \\, a^{4} m y - {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} y^{3}\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + {\\left(a^{4} {\\ell}^{4} y^{8} + 2 \\, a^{4} {\\ell}^{2} y^{6} - 4 \\, a^{4} m y^{2} + 4 \\, a^{4} m^{2} - {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} y^{4}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right)}{{\\left({\\mu_0}^{2} - 1\\right)} {\\ell}^{2} y^{4} + {\\left({\\mu_0}^{2} - 1\\right)} y^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} m} = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}\\right)} {\\ell}^{2} m - {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} {\\ell}^{4} y^{8} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} {\\ell}^{2} y^{6} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} m y^{2} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} m^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_1}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} {\\ell}^{4} y^{8} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} {\\ell}^{2} y^{6} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} m y^{2} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} m^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} a^{4} {\\beta_2}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 2 \\, {\\left(2 \\, a^{4} {\\ell}^{4} y^{7} + 3 \\, a^{4} {\\ell}^{2} y^{5} - 2 \\, a^{4} m y - {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} y^{3}\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + {\\left(a^{4} {\\ell}^{4} y^{8} + 2 \\, a^{4} {\\ell}^{2} y^{6} - 4 \\, a^{4} m y^{2} + 4 \\, a^{4} m^{2} - {\\left(4 \\, a^{4} {\\ell}^{2} m - a^{4}\\right)} y^{4}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right)}{{\\left({\\mu_0}^{2} - 1\\right)} {\\ell}^{2} y^{4} + {\\left({\\mu_0}^{2} - 1\\right)} y^{2} - 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} m} = 0$$" ], "text/plain": [ "(((Mu0^3 - Mu0)*a^4*beta1^2 - (Mu0^3 - Mu0)*a^4*beta2^2)*l^4*y^4 - 4*((Mu0^3 - Mu0)*a^4*beta1 - (Mu0^3 - Mu0)*a^4*beta2)*l^2*m - ((Mu0^3 - Mu0)*a^4*beta1^2*l^4*y^8 + 2*(Mu0^3 - Mu0)*a^4*beta1^2*l^2*y^6 - 4*(Mu0^3 - Mu0)*a^4*beta1^2*m*y^2 + 4*(Mu0^3 - Mu0)*a^4*beta1^2*m^2 - (4*(Mu0^3 - Mu0)*a^4*beta1^2*l^2*m - (Mu0^3 - Mu0)*a^4*beta1^2)*y^4)*diff(Phi_1(y), y)^2 + ((Mu0^3 - Mu0)*a^4*beta2^2*l^4*y^8 + 2*(Mu0^3 - Mu0)*a^4*beta2^2*l^2*y^6 - 4*(Mu0^3 - Mu0)*a^4*beta2^2*m*y^2 + 4*(Mu0^3 - Mu0)*a^4*beta2^2*m^2 - (4*(Mu0^3 - Mu0)*a^4*beta2^2*l^2*m - (Mu0^3 - Mu0)*a^4*beta2^2)*y^4)*diff(Psi_1(y), y)^2 + 2*(2*a^4*l^4*y^7 + 3*a^4*l^2*y^5 - 2*a^4*m*y - (4*a^4*l^2*m - a^4)*y^3)*diff(mu_1(y), y) + (a^4*l^4*y^8 + 2*a^4*l^2*y^6 - 4*a^4*m*y^2 + 4*a^4*m^2 - (4*a^4*l^2*m - a^4)*y^4)*diff(mu_1(y), y, y))/((Mu0^2 - 1)*l^2*y^4 + (Mu0^2 - 1)*y^2 - 2*(Mu0^2 - 1)*m) == 0" ] }, "execution_count": 67, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_mu1 = eqs[2]\n", "eq_mu1" ] }, { "cell_type": "code", "execution_count": 68, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "# eq_mu1.lhs().numerator().simplify_full()" ] }, { "cell_type": "code", "execution_count": 69, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "# eq_mu1.lhs().denominator().simplify_full()" ] }, { "cell_type": "code", "execution_count": 70, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "eq_mu1 = eq_mu1.lhs().numerator().simplify_full() == 0" ] }, { "cell_type": "code", "execution_count": 71, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m - {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} {\\ell}^{4} y^{8} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} {\\ell}^{2} y^{6} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} m y^{2} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} m^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} {\\ell}^{4} y^{8} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} {\\ell}^{2} y^{6} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} m y^{2} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} m^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 2 \\, {\\left(2 \\, {\\ell}^{4} y^{7} + 3 \\, {\\ell}^{2} y^{5} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{3} - 2 \\, m y\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + {\\left({\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m - {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} {\\ell}^{4} y^{8} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} {\\ell}^{2} y^{6} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} m y^{2} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} m^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right)^{2} + {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} {\\ell}^{4} y^{8} + 2 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} {\\ell}^{2} y^{6} - 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} m y^{2} + 4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} m^{2} - {\\left(4 \\, {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2} {\\ell}^{2} m - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} y^{4}\\right)} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)^{2} + 2 \\, {\\left(2 \\, {\\ell}^{4} y^{7} + 3 \\, {\\ell}^{2} y^{5} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{3} - 2 \\, m y\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + {\\left({\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right) = 0$$" ], "text/plain": [ "((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^4*y^4 - 4*((Mu0^3 - Mu0)*beta1 - (Mu0^3 - Mu0)*beta2)*l^2*m - ((Mu0^3 - Mu0)*beta1^2*l^4*y^8 + 2*(Mu0^3 - Mu0)*beta1^2*l^2*y^6 - 4*(Mu0^3 - Mu0)*beta1^2*m*y^2 + 4*(Mu0^3 - Mu0)*beta1^2*m^2 - (4*(Mu0^3 - Mu0)*beta1^2*l^2*m - (Mu0^3 - Mu0)*beta1^2)*y^4)*diff(Phi_1(y), y)^2 + ((Mu0^3 - Mu0)*beta2^2*l^4*y^8 + 2*(Mu0^3 - Mu0)*beta2^2*l^2*y^6 - 4*(Mu0^3 - Mu0)*beta2^2*m*y^2 + 4*(Mu0^3 - Mu0)*beta2^2*m^2 - (4*(Mu0^3 - Mu0)*beta2^2*l^2*m - (Mu0^3 - Mu0)*beta2^2)*y^4)*diff(Psi_1(y), y)^2 + 2*(2*l^4*y^7 + 3*l^2*y^5 - (4*l^2*m - 1)*y^3 - 2*m*y)*diff(mu_1(y), y) + (l^4*y^8 + 2*l^2*y^6 - (4*l^2*m - 1)*y^4 - 4*m*y^2 + 4*m^2)*diff(mu_1(y), y, y) == 0" ] }, "execution_count": 71, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_mu1 = (eq_mu1/a^4).simplify_full()\n", "eq_mu1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We plug the solutions obtained previously for $\\phi_1(r)$ and $\\psi_1(r)$ in this equation:" ] }, { "cell_type": "code", "execution_count": 72, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m + 2 \\, {\\left(2 \\, {\\ell}^{4} y^{7} + 3 \\, {\\ell}^{2} y^{5} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{3} - 2 \\, m y\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + {\\left({\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m + 2 \\, {\\left(2 \\, {\\ell}^{4} y^{7} + 3 \\, {\\ell}^{2} y^{5} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{3} - 2 \\, m y\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + {\\left({\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right) = 0$$" ], "text/plain": [ "((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^4*y^4 - (Mu0^3 - Mu0)*P^2*beta1^2 + (Mu0^3 - Mu0)*Q^2*beta2^2 - 4*((Mu0^3 - Mu0)*beta1 - (Mu0^3 - Mu0)*beta2)*l^2*m + 2*(2*l^4*y^7 + 3*l^2*y^5 - (4*l^2*m - 1)*y^3 - 2*m*y)*diff(mu_1(y), y) + (l^4*y^8 + 2*l^2*y^6 - (4*l^2*m - 1)*y^4 - 4*m*y^2 + 4*m^2)*diff(mu_1(y), y, y) == 0" ] }, "execution_count": 72, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_mu1 = eq_mu1.substitute_function(Phi_1, Phi1_sol).substitute_function(Psi_1, Psi1_sol)\n", "eq_mu1 = eq_mu1.simplify_full()\n", "eq_mu1" ] }, { "cell_type": "code", "execution_count": 73, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m + 2 \\, {\\left(2 \\, {\\ell}^{4} y^{7} + 3 \\, {\\ell}^{2} y^{5} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{3} - 2 \\, m y\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + {\\left({\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m + 2 \\, {\\left(2 \\, {\\ell}^{4} y^{7} + 3 \\, {\\ell}^{2} y^{5} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{3} - 2 \\, m y\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + {\\left({\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}\\right)} \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right)$$" ], "text/plain": [ "((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^4*y^4 - (Mu0^3 - Mu0)*P^2*beta1^2 + (Mu0^3 - Mu0)*Q^2*beta2^2 - 4*((Mu0^3 - Mu0)*beta1 - (Mu0^3 - Mu0)*beta2)*l^2*m + 2*(2*l^4*y^7 + 3*l^2*y^5 - (4*l^2*m - 1)*y^3 - 2*m*y)*diff(mu_1(y), y) + (l^4*y^8 + 2*l^2*y^6 - (4*l^2*m - 1)*y^4 - 4*m*y^2 + 4*m^2)*diff(mu_1(y), y, y)" ] }, "execution_count": 73, "metadata": { }, "output_type": "execute_result" } ], "source": [ "lhs = eq_mu1.lhs()\n", "lhs = lhs.simplify_full()\n", "lhs" ] }, { "cell_type": "code", "execution_count": 74, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}^{2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}^{2}$$" ], "text/plain": [ "(l^2*y^4 + y^2 - 2*m)^2" ] }, "execution_count": 74, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s = lhs.coefficient(diff(mu_1(y), y, 2)) # coefficient of mu_1''\n", "s.factor()" ] }, { "cell_type": "code", "execution_count": 75, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m + 2 \\, {\\left(2 \\, {\\ell}^{4} y^{7} + 3 \\, {\\ell}^{2} y^{5} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{3} - 2 \\, m y\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right)}{{\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m + 2 \\, {\\left(2 \\, {\\ell}^{4} y^{7} + 3 \\, {\\ell}^{2} y^{5} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{3} - 2 \\, m y\\right)} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right)}{{\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}}$$" ], "text/plain": [ "(((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^4*y^4 - (Mu0^3 - Mu0)*P^2*beta1^2 + (Mu0^3 - Mu0)*Q^2*beta2^2 - 4*((Mu0^3 - Mu0)*beta1 - (Mu0^3 - Mu0)*beta2)*l^2*m + 2*(2*l^4*y^7 + 3*l^2*y^5 - (4*l^2*m - 1)*y^3 - 2*m*y)*diff(mu_1(y), y))/(l^4*y^8 + 2*l^2*y^6 - (4*l^2*m - 1)*y^4 - 4*m*y^2 + 4*m^2)" ] }, "execution_count": 75, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = (lhs/s - diff(mu_1(y), y, 2)).simplify_full()\n", "s1" ] }, { "cell_type": "code", "execution_count": 76, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, {\\ell}^{2} y^{2} + 1\\right)} y}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, {\\ell}^{2} y^{2} + 1\\right)} y}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$$" ], "text/plain": [ "2*(2*l^2*y^2 + 1)*y/(l^2*y^4 + y^2 - 2*m)" ] }, "execution_count": 76, "metadata": { }, "output_type": "execute_result" } ], "source": [ "b1 = s1.coefficient(diff(mu_1(y), y)).factor()\n", "b1" ] }, { "cell_type": "code", "execution_count": 77, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m}{{\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{4} y^{4} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}\\right)} {\\ell}^{2} m}{{\\ell}^{4} y^{8} + 2 \\, {\\ell}^{2} y^{6} - {\\left(4 \\, {\\ell}^{2} m - 1\\right)} y^{4} - 4 \\, m y^{2} + 4 \\, m^{2}}$$" ], "text/plain": [ "(((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^4*y^4 - (Mu0^3 - Mu0)*P^2*beta1^2 + (Mu0^3 - Mu0)*Q^2*beta2^2 - 4*((Mu0^3 - Mu0)*beta1 - (Mu0^3 - Mu0)*beta2)*l^2*m)/(l^4*y^8 + 2*l^2*y^6 - (4*l^2*m - 1)*y^4 - 4*m*y^2 + 4*m^2)" ] }, "execution_count": 77, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = (s1 - b1*diff(mu_1(y), y)).simplify_full()\n", "s2" ] }, { "cell_type": "code", "execution_count": 78, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\beta_1}^{2} {\\ell}^{4} y^{4} - {\\beta_2}^{2} {\\ell}^{4} y^{4} - {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\beta_1} {\\ell}^{2} m + 4 \\, {\\beta_2} {\\ell}^{2} m\\right)} {\\left({\\mu_0} + 1\\right)} {\\left({\\mu_0} - 1\\right)} {\\mu_0}}{{\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\beta_1}^{2} {\\ell}^{4} y^{4} - {\\beta_2}^{2} {\\ell}^{4} y^{4} - {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\beta_1} {\\ell}^{2} m + 4 \\, {\\beta_2} {\\ell}^{2} m\\right)} {\\left({\\mu_0} + 1\\right)} {\\left({\\mu_0} - 1\\right)} {\\mu_0}}{{\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}^{2}}$$" ], "text/plain": [ "(beta1^2*l^4*y^4 - beta2^2*l^4*y^4 - P^2*beta1^2 + Q^2*beta2^2 - 4*beta1*l^2*m + 4*beta2*l^2*m)*(Mu0 + 1)*(Mu0 - 1)*Mu0/(l^2*y^4 + y^2 - 2*m)^2" ] }, "execution_count": 78, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2.factor()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The equation for $\\mu_1$ is thus:" ] }, { "cell_type": "code", "execution_count": 79, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\beta_1}^{2} {\\ell}^{4} y^{4} - {\\beta_2}^{2} {\\ell}^{4} y^{4} - {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\beta_1} {\\ell}^{2} m + 4 \\, {\\beta_2} {\\ell}^{2} m\\right)} {\\left({\\mu_0} + 1\\right)} {\\left({\\mu_0} - 1\\right)} {\\mu_0}}{{\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}^{2}} + \\frac{2 \\, {\\left(2 \\, {\\ell}^{2} y^{2} + 1\\right)} y \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right)}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m} + \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right) = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left({\\beta_1}^{2} {\\ell}^{4} y^{4} - {\\beta_2}^{2} {\\ell}^{4} y^{4} - {\\mathcal{P}'}^{2} {\\beta_1}^{2} + {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 4 \\, {\\beta_1} {\\ell}^{2} m + 4 \\, {\\beta_2} {\\ell}^{2} m\\right)} {\\left({\\mu_0} + 1\\right)} {\\left({\\mu_0} - 1\\right)} {\\mu_0}}{{\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}^{2}} + \\frac{2 \\, {\\left(2 \\, {\\ell}^{2} y^{2} + 1\\right)} y \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right)}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m} + \\frac{\\partial^{2}}{(\\partial y)^{2}}\\mu_{1}\\left(y\\right) = 0$$" ], "text/plain": [ "(beta1^2*l^4*y^4 - beta2^2*l^4*y^4 - P^2*beta1^2 + Q^2*beta2^2 - 4*beta1*l^2*m + 4*beta2*l^2*m)*(Mu0 + 1)*(Mu0 - 1)*Mu0/(l^2*y^4 + y^2 - 2*m)^2 + 2*(2*l^2*y^2 + 1)*y*diff(mu_1(y), y)/(l^2*y^4 + y^2 - 2*m) + diff(mu_1(y), y, y) == 0" ] }, "execution_count": 79, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_mu1 = diff(mu_1(y), y, 2) + b1*diff(mu_1(y), y) + s2.factor() == 0\n", "eq_mu1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Given that \n", "$$ \\mu_1(y) = - \\sin\\Theta_0 \\; \\theta_1(y) = - \\sqrt{1-\\mu_0^2} \\; \\theta_1(y), \\qquad \\sin2\\Theta_0 = 2\\mu_0\\sqrt{1-\\mu_0^2}$$\n", "and\n", "$$\\mathcal{P}' = \\mathcal{P}/\\beta_1^2 \\qquad\\mbox{and}\\qquad \\mathcal{Q}' = \\mathcal{Q}/\\beta_1^2,$$ \n", "we get for the equation for $\\theta_1$:\n", "$$ \\theta_1'' + \\frac{2y(2\\ell^2 y^2 + 1)}{\\ell^2 y^4 + y^2 - 2m} \\, \\theta_1' + \\frac{\\beta_2^{-2}\\mathcal{Q}^2 - \\beta_1^{-2}\\mathcal{P}^2 - 4 (\\beta_1 - \\beta_2) \\ell^2 m + (\\beta_1^2 - \\beta_2^2) \\ell^4 y^4}{2(\\ell^2 y^4 + y^2 - 2m)^2}\\sin(2\\Theta_0) = 0 $$ \n", "This agrees with Eq. (4.35) of the paper." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Solving the equation for $\\mu_1$" ] }, { "cell_type": "code", "execution_count": 80, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{2} - \\int \\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{2} y - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} \\int \\frac{{\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y^{2} + {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} - K_{1}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{2} - \\int \\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{2} y - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} \\int \\frac{{\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y^{2} + {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} - K_{1}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y}$$" ], "text/plain": [ "_K2 - integrate((((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^2*y + (Mu0^3 - Mu0)*integrate(-((beta1^2 - beta2^2)*l^2*y^2 + P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*beta1 + 2*beta2)*l^2*m)/(l^2*y^4 + y^2 - 2*m), y) - _K1)/(l^2*y^4 + y^2 - 2*m), y)" ] }, "execution_count": 80, "metadata": { }, "output_type": "execute_result" } ], "source": [ "mu1_sol(y) = desolve(eq_mu1, mu_1(y), ivar=y)\n", "mu1_sol(y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us check that `mu1_sol` is indeed a solution of the equation for $\\mu_1$:" ] }, { "cell_type": "code", "execution_count": 81, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0 = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}0 = 0$$" ], "text/plain": [ "0 == 0" ] }, "execution_count": 81, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_mu1.substitute_function(mu_1, mu1_sol).simplify_full()" ] }, { "cell_type": "code", "execution_count": 82, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{2} - \\int \\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{2} y - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} \\int \\frac{{\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y^{2} + {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} - K_{1}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{2} - \\int \\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{2} y - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} \\int \\frac{{\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y^{2} + {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} - K_{1}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y}$$" ], "text/plain": [ "_K2 - integrate((((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^2*y + (Mu0^3 - Mu0)*integrate(-((beta1^2 - beta2^2)*l^2*y^2 + P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*beta1 + 2*beta2)*l^2*m)/(l^2*y^4 + y^2 - 2*m), y) - _K1)/(l^2*y^4 + y^2 - 2*m), y)" ] }, "execution_count": 82, "metadata": { }, "output_type": "execute_result" } ], "source": [ "mu1_sol(y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The innermost integral can be written\n", "$$ (\\beta_1^2 - \\beta_2^2) \\ell^2 \\; s_1(y) + \\left({\\mathcal{P}'}^2 \\beta_1^2 - {\\mathcal{Q}'}\\beta_2^2 - 2 (\\beta_1^2-\\beta_2^2 - 2 (\\beta_1-\\beta_2))\\ell^2 m \\right) \\; s_2(y)$$\n", "with \n", "$$ s_1(y) := \\int^y \\frac{\\bar{y}^2}{\\ell^2 \\bar{y}^4 + \\bar{y}^2 - 2m} \\, \\mathrm{d}\\bar{y} \\qquad \\mbox{and}\\qquad s_2(y) := \\int^y \\frac{\\mathrm{d}\\bar{y}}{\\ell^2 \\bar{y}^4 + \\bar{y}^2 - 2m} .$$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us evaluate $s_1$ by means of FriCAS:" ] }, { "cell_type": "code", "execution_count": 83, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + y\\right) - \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(-\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + y\\right) - \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + y\\right) + \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(-\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + y\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + y\\right) - \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(-\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + y\\right) - \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + y\\right) + \\frac{1}{2} \\, \\sqrt{\\frac{1}{2}} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}} \\log\\left(-\\frac{\\sqrt{\\frac{1}{2}} {\\left(8 \\, {\\ell}^{4} m + {\\ell}^{2}\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{4} m + {\\ell}^{2}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} - 1}{8 \\, {\\ell}^{4} m + {\\ell}^{2}}}}{\\sqrt{8 \\, {\\ell}^{6} m + {\\ell}^{4}}} + y\\right)$$" ], "text/plain": [ "1/2*sqrt(1/2)*sqrt(-((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) + 1)/(8*l^4*m + l^2))*log(sqrt(1/2)*(8*l^4*m + l^2)*sqrt(-((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) + 1)/(8*l^4*m + l^2))/sqrt(8*l^6*m + l^4) + y) - 1/2*sqrt(1/2)*sqrt(-((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) + 1)/(8*l^4*m + l^2))*log(-sqrt(1/2)*(8*l^4*m + l^2)*sqrt(-((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) + 1)/(8*l^4*m + l^2))/sqrt(8*l^6*m + l^4) + y) - 1/2*sqrt(1/2)*sqrt(((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) - 1)/(8*l^4*m + l^2))*log(sqrt(1/2)*(8*l^4*m + l^2)*sqrt(((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) - 1)/(8*l^4*m + l^2))/sqrt(8*l^6*m + l^4) + y) + 1/2*sqrt(1/2)*sqrt(((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) - 1)/(8*l^4*m + l^2))*log(-sqrt(1/2)*(8*l^4*m + l^2)*sqrt(((8*l^4*m + l^2)/sqrt(8*l^6*m + l^4) - 1)/(8*l^4*m + l^2))/sqrt(8*l^6*m + l^4) + y)" ] }, "execution_count": 83, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = integrate(y^2/(l^2*y^4 + y^2 - 2*m), y, algorithm='fricas')\n", "s1" ] }, { "cell_type": "code", "execution_count": 84, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} \\log\\left(\\frac{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} y - \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} y + \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}\\right) + \\sqrt{2} \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} \\log\\left(\\frac{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} y + \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} y - \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}\\right)}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{3}{4}} {\\ell}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} \\log\\left(\\frac{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} y - \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} y + \\sqrt{8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}\\right) + \\sqrt{2} \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} \\log\\left(\\frac{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} y + \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{\\sqrt{2} {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell} y - \\sqrt{-8 \\, {\\ell}^{2} m - \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}\\right)}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{3}{4}} {\\ell}}$$" ], "text/plain": [ "1/4*(sqrt(2)*sqrt(8*l^2*m - sqrt(8*l^2*m + 1) + 1)*log((sqrt(2)*(8*l^2*m + 1)^(1/4)*l*y - sqrt(8*l^2*m - sqrt(8*l^2*m + 1) + 1))/(sqrt(2)*(8*l^2*m + 1)^(1/4)*l*y + sqrt(8*l^2*m - sqrt(8*l^2*m + 1) + 1))) + sqrt(2)*sqrt(-8*l^2*m - sqrt(8*l^2*m + 1) - 1)*log((sqrt(2)*(8*l^2*m + 1)^(1/4)*l*y + sqrt(-8*l^2*m - sqrt(8*l^2*m + 1) - 1))/(sqrt(2)*(8*l^2*m + 1)^(1/4)*l*y - sqrt(-8*l^2*m - sqrt(8*l^2*m + 1) - 1))))/((8*l^2*m + 1)^(3/4)*l)" ] }, "execution_count": 84, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = s1.canonicalize_radical().simplify_log()\n", "s1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Check:" ] }, { "cell_type": "code", "execution_count": 85, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y^{2}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y^{2}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$$" ], "text/plain": [ "y^2/(l^2*y^4 + y^2 - 2*m)" ] }, "execution_count": 85, "metadata": { }, "output_type": "execute_result" } ], "source": [ "diff(s1, y).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Similarly, we evaluate $s_2$ by means of FriCAS:" ] }, { "cell_type": "code", "execution_count": 86, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{4} \\, \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} y + \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m - \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) + \\frac{1}{4} \\, \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} y - \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m - \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) - \\frac{1}{4} \\, \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} y + \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m + \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) + \\frac{1}{4} \\, \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} y - \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m + \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right)$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{4} \\, \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} y + \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m - \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) + \\frac{1}{4} \\, \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} y - \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m - \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) - \\frac{1}{4} \\, \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} y + \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m + \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right) + \\frac{1}{4} \\, \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}} \\log\\left(2 \\, {\\ell}^{2} y - \\frac{1}{2} \\, {\\left(8 \\, {\\ell}^{2} m + \\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} + 1\\right)} \\sqrt{-\\frac{\\frac{8 \\, {\\ell}^{2} m^{2} + m}{\\sqrt{8 \\, {\\ell}^{2} m^{3} + m^{2}}} - 1}{8 \\, {\\ell}^{2} m^{2} + m}}\\right)$$" ], "text/plain": [ "-1/4*sqrt(((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)/(8*l^2*m^2 + m))*log(2*l^2*y + 1/2*(8*l^2*m - (8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)*sqrt(((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)/(8*l^2*m^2 + m))) + 1/4*sqrt(((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)/(8*l^2*m^2 + m))*log(2*l^2*y - 1/2*(8*l^2*m - (8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)*sqrt(((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)/(8*l^2*m^2 + m))) - 1/4*sqrt(-((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) - 1)/(8*l^2*m^2 + m))*log(2*l^2*y + 1/2*(8*l^2*m + (8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)*sqrt(-((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) - 1)/(8*l^2*m^2 + m))) + 1/4*sqrt(-((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) - 1)/(8*l^2*m^2 + m))*log(2*l^2*y - 1/2*(8*l^2*m + (8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) + 1)*sqrt(-((8*l^2*m^2 + m)/sqrt(8*l^2*m^3 + m^2) - 1)/(8*l^2*m^2 + m)))" ] }, "execution_count": 86, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = integrate(1/(l^2*y^4 + y^2 - 2*m), y, algorithm='fricas')\n", "s2" ] }, { "cell_type": "code", "execution_count": 87, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} \\log\\left(\\frac{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} y - \\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1\\right)}}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} y + \\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1\\right)}}\\right) + \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} \\log\\left(\\frac{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} y - \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1\\right)}}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} y + \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1\\right)}}\\right)}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{3}{4}} \\sqrt{m}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} \\log\\left(\\frac{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} y - \\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1\\right)}}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} y + \\sqrt{-8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1\\right)}}\\right) + \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} \\log\\left(\\frac{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} y - \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1\\right)}}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{1}{4}} {\\ell}^{2} \\sqrt{m} y + \\sqrt{8 \\, {\\ell}^{2} m + \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} {\\left(\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1\\right)}}\\right)}{4 \\, {\\left(8 \\, {\\ell}^{2} m + 1\\right)}^{\\frac{3}{4}} \\sqrt{m}}$$" ], "text/plain": [ "1/4*(sqrt(-8*l^2*m + sqrt(8*l^2*m + 1) - 1)*log((4*(8*l^2*m + 1)^(1/4)*l^2*sqrt(m)*y - sqrt(-8*l^2*m + sqrt(8*l^2*m + 1) - 1)*(sqrt(8*l^2*m + 1) + 1))/(4*(8*l^2*m + 1)^(1/4)*l^2*sqrt(m)*y + sqrt(-8*l^2*m + sqrt(8*l^2*m + 1) - 1)*(sqrt(8*l^2*m + 1) + 1))) + sqrt(8*l^2*m + sqrt(8*l^2*m + 1) + 1)*log((4*(8*l^2*m + 1)^(1/4)*l^2*sqrt(m)*y - sqrt(8*l^2*m + sqrt(8*l^2*m + 1) + 1)*(sqrt(8*l^2*m + 1) - 1))/(4*(8*l^2*m + 1)^(1/4)*l^2*sqrt(m)*y + sqrt(8*l^2*m + sqrt(8*l^2*m + 1) + 1)*(sqrt(8*l^2*m + 1) - 1))))/((8*l^2*m + 1)^(3/4)*sqrt(m))" ] }, "execution_count": 87, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = s2.canonicalize_radical().simplify_log()\n", "s2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Check:" ] }, { "cell_type": "code", "execution_count": 88, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$$" ], "text/plain": [ "1/(l^2*y^4 + y^2 - 2*m)" ] }, "execution_count": 88, "metadata": { }, "output_type": "execute_result" } ], "source": [ "diff(s2, y).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "In the above expressions for $s_1(y)$ and $s_2(y)$ there appears the factor \n", "$$\\mathfrak{P} = \\sqrt{1 + 8\\ell^2 m},$$\n", "which we represent by the symbolic variable `B`" ] }, { "cell_type": "code", "execution_count": 89, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "B = var('B')\n", "assume(B > 1)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us make $B$ appear in $s_1$:" ] }, { "cell_type": "code", "execution_count": 90, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} \\sqrt{B^{2} - B} \\log\\left(\\frac{\\sqrt{2} \\sqrt{B} {\\ell} y - \\sqrt{B^{2} - B}}{\\sqrt{2} \\sqrt{B} {\\ell} y + \\sqrt{B^{2} - B}}\\right) + \\sqrt{2} \\sqrt{-B^{2} - B} \\log\\left(\\frac{\\sqrt{2} \\sqrt{B} {\\ell} y + \\sqrt{-B^{2} - B}}{\\sqrt{2} \\sqrt{B} {\\ell} y - \\sqrt{-B^{2} - B}}\\right)}{4 \\, B^{\\frac{3}{2}} {\\ell}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} \\sqrt{B^{2} - B} \\log\\left(\\frac{\\sqrt{2} \\sqrt{B} {\\ell} y - \\sqrt{B^{2} - B}}{\\sqrt{2} \\sqrt{B} {\\ell} y + \\sqrt{B^{2} - B}}\\right) + \\sqrt{2} \\sqrt{-B^{2} - B} \\log\\left(\\frac{\\sqrt{2} \\sqrt{B} {\\ell} y + \\sqrt{-B^{2} - B}}{\\sqrt{2} \\sqrt{B} {\\ell} y - \\sqrt{-B^{2} - B}}\\right)}{4 \\, B^{\\frac{3}{2}} {\\ell}}$$" ], "text/plain": [ "1/4*(sqrt(2)*sqrt(B^2 - B)*log((sqrt(2)*sqrt(B)*l*y - sqrt(B^2 - B))/(sqrt(2)*sqrt(B)*l*y + sqrt(B^2 - B))) + sqrt(2)*sqrt(-B^2 - B)*log((sqrt(2)*sqrt(B)*l*y + sqrt(-B^2 - B))/(sqrt(2)*sqrt(B)*l*y - sqrt(-B^2 - B))))/(B^(3/2)*l)" ] }, "execution_count": 90, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = s1.subs({l^2: (B^2 - 1)/(8*m)}).simplify_full()\n", "s1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "In this expression, there appears the term $\\sqrt{-B^2-B}$ which is imaginary since $B>1$. \n", "We there rewrite it as $i\\sqrt{B}\\sqrt{B+1}$:" ] }, { "cell_type": "code", "execution_count": 91, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{i \\, \\sqrt{2} \\sqrt{B + 1} \\sqrt{B} \\log\\left(\\frac{\\sqrt{2} \\sqrt{B} {\\ell} y + i \\, \\sqrt{B + 1} \\sqrt{B}}{\\sqrt{2} \\sqrt{B} {\\ell} y - i \\, \\sqrt{B + 1} \\sqrt{B}}\\right) + \\sqrt{2} \\sqrt{B - 1} \\sqrt{B} \\log\\left(\\frac{\\sqrt{2} \\sqrt{B} {\\ell} y - \\sqrt{B - 1} \\sqrt{B}}{\\sqrt{2} \\sqrt{B} {\\ell} y + \\sqrt{B - 1} \\sqrt{B}}\\right)}{4 \\, B^{\\frac{3}{2}} {\\ell}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{i \\, \\sqrt{2} \\sqrt{B + 1} \\sqrt{B} \\log\\left(\\frac{\\sqrt{2} \\sqrt{B} {\\ell} y + i \\, \\sqrt{B + 1} \\sqrt{B}}{\\sqrt{2} \\sqrt{B} {\\ell} y - i \\, \\sqrt{B + 1} \\sqrt{B}}\\right) + \\sqrt{2} \\sqrt{B - 1} \\sqrt{B} \\log\\left(\\frac{\\sqrt{2} \\sqrt{B} {\\ell} y - \\sqrt{B - 1} \\sqrt{B}}{\\sqrt{2} \\sqrt{B} {\\ell} y + \\sqrt{B - 1} \\sqrt{B}}\\right)}{4 \\, B^{\\frac{3}{2}} {\\ell}}$$" ], "text/plain": [ "1/4*(I*sqrt(2)*sqrt(B + 1)*sqrt(B)*log((sqrt(2)*sqrt(B)*l*y + I*sqrt(B + 1)*sqrt(B))/(sqrt(2)*sqrt(B)*l*y - I*sqrt(B + 1)*sqrt(B))) + sqrt(2)*sqrt(B - 1)*sqrt(B)*log((sqrt(2)*sqrt(B)*l*y - sqrt(B - 1)*sqrt(B))/(sqrt(2)*sqrt(B)*l*y + sqrt(B - 1)*sqrt(B))))/(B^(3/2)*l)" ] }, "execution_count": 91, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = s1.subs({sqrt(-B^2 - B): I*sqrt(B)*sqrt(B + 1), \n", " sqrt(B^2 - B): sqrt(B)*sqrt(B - 1)})\n", "s1" ] }, { "cell_type": "code", "execution_count": 92, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{i \\, \\sqrt{2} \\sqrt{B + 1} \\log\\left(\\frac{\\sqrt{2} {\\ell} y + i \\, \\sqrt{B + 1}}{\\sqrt{2} {\\ell} y - i \\, \\sqrt{B + 1}}\\right) + \\sqrt{2} \\sqrt{B - 1} \\log\\left(\\frac{\\sqrt{2} {\\ell} y - \\sqrt{B - 1}}{\\sqrt{2} {\\ell} y + \\sqrt{B - 1}}\\right)}{4 \\, B {\\ell}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{i \\, \\sqrt{2} \\sqrt{B + 1} \\log\\left(\\frac{\\sqrt{2} {\\ell} y + i \\, \\sqrt{B + 1}}{\\sqrt{2} {\\ell} y - i \\, \\sqrt{B + 1}}\\right) + \\sqrt{2} \\sqrt{B - 1} \\log\\left(\\frac{\\sqrt{2} {\\ell} y - \\sqrt{B - 1}}{\\sqrt{2} {\\ell} y + \\sqrt{B - 1}}\\right)}{4 \\, B {\\ell}}$$" ], "text/plain": [ "1/4*(I*sqrt(2)*sqrt(B + 1)*log((sqrt(2)*l*y + I*sqrt(B + 1))/(sqrt(2)*l*y - I*sqrt(B + 1))) + sqrt(2)*sqrt(B - 1)*log((sqrt(2)*l*y - sqrt(B - 1))/(sqrt(2)*l*y + sqrt(B - 1))))/(B*l)" ] }, "execution_count": 92, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = s1.simplify_log()\n", "s1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "In the first $\\log$, we recognize the $\\mathrm{arctan}$ function, via the identity\n", "$$\n", " \\mathrm{arctan}\\, x = \\frac{i}{2} \\ln\\left( \\frac{i + x}{i - x} \\right), \n", "$$\n", "which we use in the form\n", "$$\n", "i \\ln\\left( \\frac{x + i}{x - i} \\right) = 2 \\mathrm{arctan}(x) - \\pi\n", "$$\n", "as we can check:" ] }, { "cell_type": "code", "execution_count": 93, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}0$$" ], "text/plain": [ "0" ] }, "execution_count": 93, "metadata": { }, "output_type": "execute_result" } ], "source": [ "taylor(I*ln((x+I)/(x-I)) - 2*atan(x) + pi, x, 0, 10)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Thus, we set, disregarding the additive constant $-\\pi$," ] }, { "cell_type": "code", "execution_count": 94, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} {\\left(2 \\, \\sqrt{B + 1} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B + 1}}\\right) + \\sqrt{B - 1} \\log\\left(\\frac{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} - 1}{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} + 1}\\right)\\right)}}{4 \\, B {\\ell}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{2} {\\left(2 \\, \\sqrt{B + 1} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B + 1}}\\right) + \\sqrt{B - 1} \\log\\left(\\frac{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} - 1}{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} + 1}\\right)\\right)}}{4 \\, B {\\ell}}$$" ], "text/plain": [ "1/4*sqrt(2)*(2*sqrt(B + 1)*arctan(sqrt(2)*l*y/sqrt(B + 1)) + sqrt(B - 1)*log((sqrt(2)*l*y/sqrt(B - 1) - 1)/(sqrt(2)*l*y/sqrt(B - 1) + 1)))/(B*l)" ] }, "execution_count": 94, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s1 = sqrt(2)/(4*B*l)*(2*sqrt(B+1)*atan(sqrt(2)*l/sqrt(B+1)*y)\n", " + sqrt(B-1)*ln((sqrt(2)*l/sqrt(B-1)*y - 1)/(sqrt(2)*l/sqrt(B-1)*y + 1)))\n", "s1" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us check that we have indeed a primitive of $y\\mapsto \\frac{y^2}{\\ell^2 y^4 + y^2 - 2m}$:" ] }, { "cell_type": "code", "execution_count": 95, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{4 \\, {\\ell}^{2} y^{2}}{4 \\, {\\ell}^{4} y^{4} + 4 \\, {\\ell}^{2} y^{2} - B^{2} + 1}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{4 \\, {\\ell}^{2} y^{2}}{4 \\, {\\ell}^{4} y^{4} + 4 \\, {\\ell}^{2} y^{2} - B^{2} + 1}$$" ], "text/plain": [ "4*l^2*y^2/(4*l^4*y^4 + 4*l^2*y^2 - B^2 + 1)" ] }, "execution_count": 95, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Ds1 = diff(s1, y).simplify_full()\n", "Ds1" ] }, { "cell_type": "code", "execution_count": 96, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y^{2}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y^{2}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$$" ], "text/plain": [ "y^2/(l^2*y^4 + y^2 - 2*m)" ] }, "execution_count": 96, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Ds1.subs({B: sqrt(1 + 8*l^2*m)}).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Similarly, we can express $s_2$ in terms of $B$:" ] }, { "cell_type": "code", "execution_count": 97, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{B^{2} + B} \\log\\left(\\frac{{\\left(B + 1\\right)} \\sqrt{B} \\sqrt{m} y - 2 \\, \\sqrt{B^{2} + B} m}{{\\left(B + 1\\right)} \\sqrt{B} \\sqrt{m} y + 2 \\, \\sqrt{B^{2} + B} m}\\right) + \\sqrt{-B^{2} + B} \\log\\left(\\frac{{\\left(B - 1\\right)} \\sqrt{B} \\sqrt{m} y - 2 \\, \\sqrt{-B^{2} + B} m}{{\\left(B - 1\\right)} \\sqrt{B} \\sqrt{m} y + 2 \\, \\sqrt{-B^{2} + B} m}\\right)}{4 \\, B^{\\frac{3}{2}} \\sqrt{m}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{B^{2} + B} \\log\\left(\\frac{{\\left(B + 1\\right)} \\sqrt{B} \\sqrt{m} y - 2 \\, \\sqrt{B^{2} + B} m}{{\\left(B + 1\\right)} \\sqrt{B} \\sqrt{m} y + 2 \\, \\sqrt{B^{2} + B} m}\\right) + \\sqrt{-B^{2} + B} \\log\\left(\\frac{{\\left(B - 1\\right)} \\sqrt{B} \\sqrt{m} y - 2 \\, \\sqrt{-B^{2} + B} m}{{\\left(B - 1\\right)} \\sqrt{B} \\sqrt{m} y + 2 \\, \\sqrt{-B^{2} + B} m}\\right)}{4 \\, B^{\\frac{3}{2}} \\sqrt{m}}$$" ], "text/plain": [ "1/4*(sqrt(B^2 + B)*log(((B + 1)*sqrt(B)*sqrt(m)*y - 2*sqrt(B^2 + B)*m)/((B + 1)*sqrt(B)*sqrt(m)*y + 2*sqrt(B^2 + B)*m)) + sqrt(-B^2 + B)*log(((B - 1)*sqrt(B)*sqrt(m)*y - 2*sqrt(-B^2 + B)*m)/((B - 1)*sqrt(B)*sqrt(m)*y + 2*sqrt(-B^2 + B)*m)))/(B^(3/2)*sqrt(m))" ] }, "execution_count": 97, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = s2.subs({l^2: (B^2 - 1)/(8*m)}).simplify_full()\n", "s2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Since $B>1$, we replace $\\sqrt{-B^2 + B}$ by $i\\sqrt{B}\\sqrt{B-1}$:" ] }, { "cell_type": "code", "execution_count": 98, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{B + 1} \\sqrt{B} \\log\\left(\\frac{{\\left(B + 1\\right)} \\sqrt{B} \\sqrt{m} y - 2 \\, \\sqrt{B + 1} \\sqrt{B} m}{{\\left(B + 1\\right)} \\sqrt{B} \\sqrt{m} y + 2 \\, \\sqrt{B + 1} \\sqrt{B} m}\\right) + i \\, \\sqrt{B - 1} \\sqrt{B} \\log\\left(\\frac{{\\left(B - 1\\right)} \\sqrt{B} \\sqrt{m} y - 2 i \\, \\sqrt{B - 1} \\sqrt{B} m}{{\\left(B - 1\\right)} \\sqrt{B} \\sqrt{m} y + 2 i \\, \\sqrt{B - 1} \\sqrt{B} m}\\right)}{4 \\, B^{\\frac{3}{2}} \\sqrt{m}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{B + 1} \\sqrt{B} \\log\\left(\\frac{{\\left(B + 1\\right)} \\sqrt{B} \\sqrt{m} y - 2 \\, \\sqrt{B + 1} \\sqrt{B} m}{{\\left(B + 1\\right)} \\sqrt{B} \\sqrt{m} y + 2 \\, \\sqrt{B + 1} \\sqrt{B} m}\\right) + i \\, \\sqrt{B - 1} \\sqrt{B} \\log\\left(\\frac{{\\left(B - 1\\right)} \\sqrt{B} \\sqrt{m} y - 2 i \\, \\sqrt{B - 1} \\sqrt{B} m}{{\\left(B - 1\\right)} \\sqrt{B} \\sqrt{m} y + 2 i \\, \\sqrt{B - 1} \\sqrt{B} m}\\right)}{4 \\, B^{\\frac{3}{2}} \\sqrt{m}}$$" ], "text/plain": [ "1/4*(sqrt(B + 1)*sqrt(B)*log(((B + 1)*sqrt(B)*sqrt(m)*y - 2*sqrt(B + 1)*sqrt(B)*m)/((B + 1)*sqrt(B)*sqrt(m)*y + 2*sqrt(B + 1)*sqrt(B)*m)) + I*sqrt(B - 1)*sqrt(B)*log(((B - 1)*sqrt(B)*sqrt(m)*y - 2*I*sqrt(B - 1)*sqrt(B)*m)/((B - 1)*sqrt(B)*sqrt(m)*y + 2*I*sqrt(B - 1)*sqrt(B)*m)))/(B^(3/2)*sqrt(m))" ] }, "execution_count": 98, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = s2.subs({sqrt(-B^2 + B): I*sqrt(B)*sqrt(B - 1), \n", " sqrt(B^2 + B): sqrt(B)*sqrt(B + 1)})\n", "s2" ] }, { "cell_type": "code", "execution_count": 99, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{B + 1} \\log\\left(\\frac{{\\left(B + 1\\right)} \\sqrt{m} y - 2 \\, \\sqrt{B + 1} m}{{\\left(B + 1\\right)} \\sqrt{m} y + 2 \\, \\sqrt{B + 1} m}\\right) + i \\, \\sqrt{B - 1} \\log\\left(\\frac{{\\left(B - 1\\right)} \\sqrt{m} y - 2 i \\, \\sqrt{B - 1} m}{{\\left(B - 1\\right)} \\sqrt{m} y + 2 i \\, \\sqrt{B - 1} m}\\right)}{4 \\, B \\sqrt{m}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sqrt{B + 1} \\log\\left(\\frac{{\\left(B + 1\\right)} \\sqrt{m} y - 2 \\, \\sqrt{B + 1} m}{{\\left(B + 1\\right)} \\sqrt{m} y + 2 \\, \\sqrt{B + 1} m}\\right) + i \\, \\sqrt{B - 1} \\log\\left(\\frac{{\\left(B - 1\\right)} \\sqrt{m} y - 2 i \\, \\sqrt{B - 1} m}{{\\left(B - 1\\right)} \\sqrt{m} y + 2 i \\, \\sqrt{B - 1} m}\\right)}{4 \\, B \\sqrt{m}}$$" ], "text/plain": [ "1/4*(sqrt(B + 1)*log(((B + 1)*sqrt(m)*y - 2*sqrt(B + 1)*m)/((B + 1)*sqrt(m)*y + 2*sqrt(B + 1)*m)) + I*sqrt(B - 1)*log(((B - 1)*sqrt(m)*y - 2*I*sqrt(B - 1)*m)/((B - 1)*sqrt(m)*y + 2*I*sqrt(B - 1)*m)))/(B*sqrt(m))" ] }, "execution_count": 99, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = s2.simplify_log()\n", "s2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Again, we use the identity\n", "$$\n", "i \\ln\\left( \\frac{x + i}{x - i} \\right) = 2 \\mathrm{arctan}(x) - \\pi\n", "$$\n", "to rewrite $s_2$ as" ] }, { "cell_type": "code", "execution_count": 100, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)}{4 \\, B \\sqrt{m}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)}{4 \\, B \\sqrt{m}}$$" ], "text/plain": [ "-1/4*(2*sqrt(B - 1)*arctan(1/2*sqrt(B - 1)*y/sqrt(m)) - sqrt(B + 1)*log((sqrt(B + 1)*y/sqrt(m) - 2)/(sqrt(B + 1)*y/sqrt(m) + 2)))/(B*sqrt(m))" ] }, "execution_count": 100, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s2 = 1/(4*B*sqrt(m))*(sqrt(B+1)*ln( (sqrt(B+1)/(2*sqrt(m))*y - 1)\n", " /(sqrt(B+1)/(2*sqrt(m))*y + 1) )\n", " - 2*sqrt(B-1)*atan(sqrt(B-1)/(2*sqrt(m))*y))\n", "s2" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us check that we have indeed a primitive of $y\\mapsto \\frac{1}{\\ell^2 y^4 + y^2 - 2m}$:" ] }, { "cell_type": "code", "execution_count": 101, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{8 \\, m}{{\\left(B^{2} - 1\\right)} y^{4} + 8 \\, m y^{2} - 16 \\, m^{2}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{8 \\, m}{{\\left(B^{2} - 1\\right)} y^{4} + 8 \\, m y^{2} - 16 \\, m^{2}}$$" ], "text/plain": [ "8*m/((B^2 - 1)*y^4 + 8*m*y^2 - 16*m^2)" ] }, "execution_count": 101, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Ds2 = diff(s2, y).simplify_full()\n", "Ds2" ] }, { "cell_type": "code", "execution_count": 102, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}$$" ], "text/plain": [ "1/(l^2*y^4 + y^2 - 2*m)" ] }, "execution_count": 102, "metadata": { }, "output_type": "execute_result" } ], "source": [ "Ds2.subs({B: sqrt(1 + 8*l^2*m)}).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Given the above expressions for $s_1(y)$ and $s_2(y)$ we rewrite the solution" ] }, { "cell_type": "code", "execution_count": 103, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{2} - \\int \\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{2} y - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} \\int \\frac{{\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y^{2} + {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} - K_{1}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}K_{2} - \\int \\frac{{\\left({\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_1}^{2} - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} {\\ell}^{2} y - {\\left({\\mu_0}^{3} - {\\mu_0}\\right)} \\int \\frac{{\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y^{2} + {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y} - K_{1}}{{\\ell}^{2} y^{4} + y^{2} - 2 \\, m}\\,{d y}$$" ], "text/plain": [ "_K2 - integrate((((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2)*l^2*y + (Mu0^3 - Mu0)*integrate(-((beta1^2 - beta2^2)*l^2*y^2 + P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*beta1 + 2*beta2)*l^2*m)/(l^2*y^4 + y^2 - 2*m), y) - _K1)/(l^2*y^4 + y^2 - 2*m), y)" ] }, "execution_count": 103, "metadata": { }, "output_type": "execute_result" } ], "source": [ "mu1_sol(y)" ] }, { "cell_type": "code", "execution_count": 104, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{C_{2}}{\\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0}} + \\frac{{\\left(2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)\\right)} C_{1}}{4 \\, \\sqrt{-{\\mu_0}^{2} + 1} B {\\mu_0} \\sqrt{m}} + \\int \\frac{4 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y - \\frac{\\sqrt{2} {\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\left(2 \\, \\sqrt{B + 1} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B + 1}}\\right) + \\sqrt{B - 1} \\log\\left(\\frac{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} - 1}{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} + 1}\\right)\\right)} {\\ell}}{B} + \\frac{{\\left({\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m\\right)} {\\left(2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)\\right)}}{B \\sqrt{m}}}{4 \\, {\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}}\\,{d y}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{C_{2}}{\\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0}} + \\frac{{\\left(2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)\\right)} C_{1}}{4 \\, \\sqrt{-{\\mu_0}^{2} + 1} B {\\mu_0} \\sqrt{m}} + \\int \\frac{4 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y - \\frac{\\sqrt{2} {\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\left(2 \\, \\sqrt{B + 1} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B + 1}}\\right) + \\sqrt{B - 1} \\log\\left(\\frac{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} - 1}{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} + 1}\\right)\\right)} {\\ell}}{B} + \\frac{{\\left({\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m\\right)} {\\left(2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)\\right)}}{B \\sqrt{m}}}{4 \\, {\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}}\\,{d y}$$" ], "text/plain": [ "-C_2/(sqrt(-Mu0^2 + 1)*Mu0) + 1/4*(2*sqrt(B - 1)*arctan(1/2*sqrt(B - 1)*y/sqrt(m)) - sqrt(B + 1)*log((sqrt(B + 1)*y/sqrt(m) - 2)/(sqrt(B + 1)*y/sqrt(m) + 2)))*C_1/(sqrt(-Mu0^2 + 1)*B*Mu0*sqrt(m)) + integrate(1/4*(4*(beta1^2 - beta2^2)*l^2*y - sqrt(2)*(beta1^2 - beta2^2)*(2*sqrt(B + 1)*arctan(sqrt(2)*l*y/sqrt(B + 1)) + sqrt(B - 1)*log((sqrt(2)*l*y/sqrt(B - 1) - 1)/(sqrt(2)*l*y/sqrt(B - 1) + 1)))*l/B + (P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*beta1 + 2*beta2)*l^2*m)*(2*sqrt(B - 1)*arctan(1/2*sqrt(B - 1)*y/sqrt(m)) - sqrt(B + 1)*log((sqrt(B + 1)*y/sqrt(m) - 2)/(sqrt(B + 1)*y/sqrt(m) + 2)))/(B*sqrt(m)))/(l^2*y^4 + y^2 - 2*m), y)" ] }, "execution_count": 104, "metadata": { }, "output_type": "execute_result" } ], "source": [ "C1, C2 = var('C_1', 'C_2') \n", "# mu1 / mu0(1-mu0^2) : \n", "mu1s0 = - C2/(Mu0*sqrt(1-Mu0^2)) - C1/(Mu0*sqrt(1-Mu0^2))*s2 \\\n", " + integrate(((beta1^2 - beta2^2)*l^2*y \n", " - (beta1^2 - beta2^2)*l^2 * s1\n", " - (P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*(beta1-beta2))*l^2*m) * s2)\n", " / (l^2*y^4 + y^2 - 2*m), \n", " y, hold=True)\n", "mu1s0" ] }, { "cell_type": "code", "execution_count": 105, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{4} \\, {\\left({\\mu_0}^{2} - 1\\right)} {\\mu_0} {\\left(\\frac{4 \\, C_{2}}{\\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0}} - \\frac{{\\left(2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)\\right)} C_{1}}{\\sqrt{-{\\mu_0}^{2} + 1} B {\\mu_0} \\sqrt{m}} - 4 \\, \\int \\frac{4 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y - \\frac{\\sqrt{2} {\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\left(2 \\, \\sqrt{B + 1} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B + 1}}\\right) + \\sqrt{B - 1} \\log\\left(\\frac{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} - 1}{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} + 1}\\right)\\right)} {\\ell}}{B} + \\frac{{\\left({\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m\\right)} {\\left(2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)\\right)}}{B \\sqrt{m}}}{4 \\, {\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}}\\,{d y}\\right)}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{4} \\, {\\left({\\mu_0}^{2} - 1\\right)} {\\mu_0} {\\left(\\frac{4 \\, C_{2}}{\\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0}} - \\frac{{\\left(2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)\\right)} C_{1}}{\\sqrt{-{\\mu_0}^{2} + 1} B {\\mu_0} \\sqrt{m}} - 4 \\, \\int \\frac{4 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y - \\frac{\\sqrt{2} {\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\left(2 \\, \\sqrt{B + 1} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B + 1}}\\right) + \\sqrt{B - 1} \\log\\left(\\frac{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} - 1}{\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B - 1}} + 1}\\right)\\right)} {\\ell}}{B} + \\frac{{\\left({\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m\\right)} {\\left(2 \\, \\sqrt{B - 1} \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - \\sqrt{B + 1} \\log\\left(\\frac{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} - 2}{\\frac{\\sqrt{B + 1} y}{\\sqrt{m}} + 2}\\right)\\right)}}{B \\sqrt{m}}}{4 \\, {\\left({\\ell}^{2} y^{4} + y^{2} - 2 \\, m\\right)}}\\,{d y}\\right)}$$" ], "text/plain": [ "1/4*(Mu0^2 - 1)*Mu0*(4*C_2/(sqrt(-Mu0^2 + 1)*Mu0) - (2*sqrt(B - 1)*arctan(1/2*sqrt(B - 1)*y/sqrt(m)) - sqrt(B + 1)*log((sqrt(B + 1)*y/sqrt(m) - 2)/(sqrt(B + 1)*y/sqrt(m) + 2)))*C_1/(sqrt(-Mu0^2 + 1)*B*Mu0*sqrt(m)) - 4*integrate(1/4*(4*(beta1^2 - beta2^2)*l^2*y - sqrt(2)*(beta1^2 - beta2^2)*(2*sqrt(B + 1)*arctan(sqrt(2)*l*y/sqrt(B + 1)) + sqrt(B - 1)*log((sqrt(2)*l*y/sqrt(B - 1) - 1)/(sqrt(2)*l*y/sqrt(B - 1) + 1)))*l/B + (P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*beta1 + 2*beta2)*l^2*m)*(2*sqrt(B - 1)*arctan(1/2*sqrt(B - 1)*y/sqrt(m)) - sqrt(B + 1)*log((sqrt(B + 1)*y/sqrt(m) - 2)/(sqrt(B + 1)*y/sqrt(m) + 2)))/(B*sqrt(m)))/(l^2*y^4 + y^2 - 2*m), y))" ] }, "execution_count": 105, "metadata": { }, "output_type": "execute_result" } ], "source": [ "mu1_sol(y) = mu1s0 * Mu0*(1-Mu0^2)\n", "mu1_sol(y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us check that we do have a solution of the equation for $\\mu_1$:" ] }, { "cell_type": "code", "execution_count": 106, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0 = 0$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}0 = 0$$" ], "text/plain": [ "0 == 0" ] }, "execution_count": 106, "metadata": { }, "output_type": "execute_result" } ], "source": [ "eq_mu1.substitute_function(mu_1, mu1_sol).simplify_full().subs({B: sqrt(1 + 8*l^2*m)}).simplify_full()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Conjugate momenta" ] }, { "cell_type": "code", "execution_count": 107, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def conjugate_momenta(lagr, qs, var):\n", " r\"\"\"\n", " Compute the conjugate momenta from a given Lagrangian.\n", "\n", " INPUT:\n", "\n", " - ``lagr`` -- symbolic expression representing the Lagrangian density\n", " - ``qs`` -- either a single symbolic function or a list/tuple of\n", " symbolic functions, representing the `q`'s; these functions must\n", " appear in ``lagr`` up to at most their first derivatives\n", " - ``var`` -- either a single variable, typically `t` (1-dimensional\n", " problem) or a list/tuple of symbolic variables; in the latter case the\n", " time coordinate must the first one\n", "\n", " OUTPUT:\n", "\n", " - list of conjugate momenta; if only one function is involved, the\n", " single conjugate momentum is returned instead.\n", "\n", " \"\"\"\n", " if not isinstance(qs, (list, tuple)):\n", " qs = [qs]\n", " if not isinstance(var, (list, tuple)):\n", " var = [var]\n", " n = len(qs)\n", " d = len(var)\n", " dqvt = [SR.var('qxxxx{}_t'.format(q)) for q in qs]\n", " subs = {diff(qs[i](*var), var[0]): dqvt[i] for i in range(n)}\n", " subs_inv = {dqvt[i]: diff(qs[i](*var), var[0]) for i in range(n)}\n", " lg = lagr.substitute(subs)\n", " ps = [diff(lg, dotq).simplify_full().substitute(subs_inv) for dotq in dqvt]\n", " if n == 1:\n", " return ps[0]\n", " return ps" ] }, { "cell_type": "code", "execution_count": 108, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-{\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right), {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{4} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y^{2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)\\right]$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-{\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} {\\ell}^{2} y^{4} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) - {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} y^{2} \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right) + 2 \\, {\\left({\\mu_0}^{2} - 1\\right)} a^{2} {\\beta_1}^{2} m \\frac{\\partial}{\\partial y}\\Phi_{1}\\left(y\\right), {\\mu_0}^{2} a^{2} {\\beta_2}^{2} {\\ell}^{2} y^{4} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) + {\\mu_0}^{2} a^{2} {\\beta_2}^{2} y^{2} \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right) - 2 \\, {\\mu_0}^{2} a^{2} {\\beta_2}^{2} m \\frac{\\partial}{\\partial y}\\Psi_{1}\\left(y\\right)\\right]$$" ], "text/plain": [ "[-(Mu0^2 - 1)*a^2*beta1^2*l^2*y^4*diff(Phi_1(y), y) - (Mu0^2 - 1)*a^2*beta1^2*y^2*diff(Phi_1(y), y) + 2*(Mu0^2 - 1)*a^2*beta1^2*m*diff(Phi_1(y), y),\n", " Mu0^2*a^2*beta2^2*l^2*y^4*diff(Psi_1(y), y) + Mu0^2*a^2*beta2^2*y^2*diff(Psi_1(y), y) - 2*Mu0^2*a^2*beta2^2*m*diff(Psi_1(y), y)]" ] }, "execution_count": 108, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pis = conjugate_momenta(L_a2, [Phi_1, Psi_1], y)\n", "pis" ] }, { "cell_type": "code", "execution_count": 109, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left({\\mu_0}^{2} - 1\\right)} {\\mathcal{P}'} a {\\beta_1}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left({\\mu_0}^{2} - 1\\right)} {\\mathcal{P}'} a {\\beta_1}$$" ], "text/plain": [ "-(Mu0^2 - 1)*P*a*beta1" ] }, "execution_count": 109, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pi_phi_y = (pis[0]/(a*beta1)).substitute_function(Phi_1, Phi1_sol).simplify_full()\n", "pi_phi_y" ] }, { "cell_type": "code", "execution_count": 110, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mu_0}^{2} {\\mathcal{Q}'} a {\\beta_2}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\mu_0}^{2} {\\mathcal{Q}'} a {\\beta_2}$$" ], "text/plain": [ "Mu0^2*Q*a*beta2" ] }, "execution_count": 110, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pi_psi_y = (pis[1]/(a*beta2)).substitute_function(Psi_1, Psi1_sol).simplify_full()\n", "pi_psi_y" ] }, { "cell_type": "code", "execution_count": 111, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "pis4 = conjugate_momenta(L_a4, [Phi_1, Psi_1, mu_1], y)" ] }, { "cell_type": "code", "execution_count": 112, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a^{4} {\\ell}^{2} y^{4} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + a^{4} y^{2} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) - 2 \\, a^{4} m \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right)}{{\\mu_0}^{2} - 1}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{a^{4} {\\ell}^{2} y^{4} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) + a^{4} y^{2} \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right) - 2 \\, a^{4} m \\frac{\\partial}{\\partial y}\\mu_{1}\\left(y\\right)}{{\\mu_0}^{2} - 1}$$" ], "text/plain": [ "-(a^4*l^2*y^4*diff(mu_1(y), y) + a^4*y^2*diff(mu_1(y), y) - 2*a^4*m*diff(mu_1(y), y))/(Mu0^2 - 1)" ] }, "execution_count": 112, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pis4[2]" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The quantity $\\pi_\\theta^y / (a^2 \\sin\\Theta_0\\cos\\Theta_0)$:" ] }, { "cell_type": "code", "execution_count": 113, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(2 \\, {\\left(\\sqrt{2} {\\mu_0} {\\beta_1}^{2} - \\sqrt{2} {\\mu_0} {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}\\right) + {\\left({\\mu_0} {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mu_0} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\mu_0} {\\beta_1}^{2} - {\\mu_0} {\\beta_2}^{2} - 2 \\, {\\mu_0} {\\beta_1} + 2 \\, {\\mu_0} {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\log\\left(\\frac{\\sqrt{m} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - 2 \\, m}{\\sqrt{m} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} + 2 \\, m}\\right)\\right)} \\sqrt{-{\\mu_0}^{2} + 1} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} + {\\left({\\left(\\sqrt{2} {\\mu_0} {\\beta_1}^{2} - \\sqrt{2} {\\mu_0} {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\log\\left(\\frac{\\sqrt{2} {\\ell} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}{\\sqrt{2} {\\ell} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}\\right) - 2 \\, {\\left({\\mu_0} {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mu_0} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\mu_0} {\\beta_1}^{2} - {\\mu_0} {\\beta_2}^{2} - 2 \\, {\\mu_0} {\\beta_1} + 2 \\, {\\mu_0} {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\arctan\\left(\\frac{y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{2 \\, \\sqrt{m}}\\right)\\right)} \\sqrt{-{\\mu_0}^{2} + 1} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} - 4 \\, {\\left({\\left({\\mu_0} {\\beta_1}^{2} - {\\mu_0} {\\beta_2}^{2}\\right)} \\sqrt{-{\\mu_0}^{2} + 1} {\\ell}^{2} \\sqrt{m} y - C_{1} \\sqrt{m}\\right)} \\sqrt{8 \\, {\\ell}^{2} m + 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0} \\sqrt{m}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(2 \\, {\\left(\\sqrt{2} {\\mu_0} {\\beta_1}^{2} - \\sqrt{2} {\\mu_0} {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}\\right) + {\\left({\\mu_0} {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mu_0} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\mu_0} {\\beta_1}^{2} - {\\mu_0} {\\beta_2}^{2} - 2 \\, {\\mu_0} {\\beta_1} + 2 \\, {\\mu_0} {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\log\\left(\\frac{\\sqrt{m} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - 2 \\, m}{\\sqrt{m} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} + 2 \\, m}\\right)\\right)} \\sqrt{-{\\mu_0}^{2} + 1} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} + {\\left({\\left(\\sqrt{2} {\\mu_0} {\\beta_1}^{2} - \\sqrt{2} {\\mu_0} {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\log\\left(\\frac{\\sqrt{2} {\\ell} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}{\\sqrt{2} {\\ell} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}\\right) - 2 \\, {\\left({\\mu_0} {\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mu_0} {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\mu_0} {\\beta_1}^{2} - {\\mu_0} {\\beta_2}^{2} - 2 \\, {\\mu_0} {\\beta_1} + 2 \\, {\\mu_0} {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\arctan\\left(\\frac{y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{2 \\, \\sqrt{m}}\\right)\\right)} \\sqrt{-{\\mu_0}^{2} + 1} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} - 4 \\, {\\left({\\left({\\mu_0} {\\beta_1}^{2} - {\\mu_0} {\\beta_2}^{2}\\right)} \\sqrt{-{\\mu_0}^{2} + 1} {\\ell}^{2} \\sqrt{m} y - C_{1} \\sqrt{m}\\right)} \\sqrt{8 \\, {\\ell}^{2} m + 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0} \\sqrt{m}}$$" ], "text/plain": [ "1/4*((2*(sqrt(2)*Mu0*beta1^2 - sqrt(2)*Mu0*beta2^2)*l*sqrt(m)*arctan(sqrt(2)*l*y/sqrt(sqrt(8*l^2*m + 1) + 1)) + (Mu0*P^2*beta1^2 - Mu0*Q^2*beta2^2 - 2*(Mu0*beta1^2 - Mu0*beta2^2 - 2*Mu0*beta1 + 2*Mu0*beta2)*l^2*m)*log((sqrt(m)*y*sqrt(sqrt(8*l^2*m + 1) + 1) - 2*m)/(sqrt(m)*y*sqrt(sqrt(8*l^2*m + 1) + 1) + 2*m)))*sqrt(-Mu0^2 + 1)*sqrt(sqrt(8*l^2*m + 1) + 1) + ((sqrt(2)*Mu0*beta1^2 - sqrt(2)*Mu0*beta2^2)*l*sqrt(m)*log((sqrt(2)*l*y*sqrt(sqrt(8*l^2*m + 1) - 1) - sqrt(8*l^2*m + 1) + 1)/(sqrt(2)*l*y*sqrt(sqrt(8*l^2*m + 1) - 1) + sqrt(8*l^2*m + 1) - 1)) - 2*(Mu0*P^2*beta1^2 - Mu0*Q^2*beta2^2 - 2*(Mu0*beta1^2 - Mu0*beta2^2 - 2*Mu0*beta1 + 2*Mu0*beta2)*l^2*m)*arctan(1/2*y*sqrt(sqrt(8*l^2*m + 1) - 1)/sqrt(m)))*sqrt(-Mu0^2 + 1)*sqrt(sqrt(8*l^2*m + 1) - 1) - 4*((Mu0*beta1^2 - Mu0*beta2^2)*sqrt(-Mu0^2 + 1)*l^2*sqrt(m)*y - C_1*sqrt(m))*sqrt(8*l^2*m + 1))/(sqrt(8*l^2*m + 1)*sqrt(-Mu0^2 + 1)*Mu0*sqrt(m))" ] }, "execution_count": 113, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pi_theta_y_a2sT0 = (- pis4[2] / (a^4*Mu0)).substitute_function(mu_1, mu1_sol).simplify_full()\n", "pi_theta_y_a2sT0 = pi_theta_y_a2sT0.subs({B: sqrt(1 + 8*l^2*m)}).simplify_full()\n", "pi_theta_y_a2sT0" ] }, { "cell_type": "code", "execution_count": 114, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{16 \\, B C_{1} m^{\\frac{3}{2}} - {\\left(2 \\, {\\left(2 \\, {\\left(B^{2} - 1\\right)} {\\mu_0} {\\beta_1} + {\\left(4 \\, {\\mu_0} {\\mathcal{P}'}^{2} - {\\left(B^{2} - 1\\right)} {\\mu_0}\\right)} {\\beta_1}^{2} - 2 \\, {\\left(B^{2} - 1\\right)} {\\mu_0} {\\beta_2} - {\\left(4 \\, {\\mu_0} {\\mathcal{Q}'}^{2} - {\\left(B^{2} - 1\\right)} {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} \\sqrt{B - 1} m \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - {\\left(2 \\, {\\left(B^{2} - 1\\right)} {\\mu_0} {\\beta_1} + {\\left(4 \\, {\\mu_0} {\\mathcal{P}'}^{2} - {\\left(B^{2} - 1\\right)} {\\mu_0}\\right)} {\\beta_1}^{2} - 2 \\, {\\left(B^{2} - 1\\right)} {\\mu_0} {\\beta_2} - {\\left(4 \\, {\\mu_0} {\\mathcal{Q}'}^{2} - {\\left(B^{2} - 1\\right)} {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} \\sqrt{B + 1} m \\log\\left(\\frac{\\sqrt{B + 1} \\sqrt{m} y - 2 \\, m}{\\sqrt{B + 1} \\sqrt{m} y + 2 \\, m}\\right) - 2 \\, {\\left(4 \\, {\\left(\\sqrt{2} {\\mu_0} {\\beta_1}^{2} - \\sqrt{2} {\\mu_0} {\\beta_2}^{2}\\right)} \\sqrt{B + 1} {\\ell} m \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B + 1}}\\right) + 2 \\, {\\left(\\sqrt{2} {\\mu_0} {\\beta_1}^{2} - \\sqrt{2} {\\mu_0} {\\beta_2}^{2}\\right)} \\sqrt{B - 1} {\\ell} m \\log\\left(\\frac{\\sqrt{2} \\sqrt{B - 1} {\\ell} y - B + 1}{\\sqrt{2} \\sqrt{B - 1} {\\ell} y + B - 1}\\right) - {\\left({\\left(B^{3} - B\\right)} {\\mu_0} {\\beta_1}^{2} - {\\left(B^{3} - B\\right)} {\\mu_0} {\\beta_2}^{2}\\right)} y\\right)} \\sqrt{m}\\right)} \\sqrt{-{\\mu_0}^{2} + 1}}{4 \\, m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{16 \\, B C_{1} m^{\\frac{3}{2}} - {\\left(2 \\, {\\left(2 \\, {\\left(B^{2} - 1\\right)} {\\mu_0} {\\beta_1} + {\\left(4 \\, {\\mu_0} {\\mathcal{P}'}^{2} - {\\left(B^{2} - 1\\right)} {\\mu_0}\\right)} {\\beta_1}^{2} - 2 \\, {\\left(B^{2} - 1\\right)} {\\mu_0} {\\beta_2} - {\\left(4 \\, {\\mu_0} {\\mathcal{Q}'}^{2} - {\\left(B^{2} - 1\\right)} {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} \\sqrt{B - 1} m \\arctan\\left(\\frac{\\sqrt{B - 1} y}{2 \\, \\sqrt{m}}\\right) - {\\left(2 \\, {\\left(B^{2} - 1\\right)} {\\mu_0} {\\beta_1} + {\\left(4 \\, {\\mu_0} {\\mathcal{P}'}^{2} - {\\left(B^{2} - 1\\right)} {\\mu_0}\\right)} {\\beta_1}^{2} - 2 \\, {\\left(B^{2} - 1\\right)} {\\mu_0} {\\beta_2} - {\\left(4 \\, {\\mu_0} {\\mathcal{Q}'}^{2} - {\\left(B^{2} - 1\\right)} {\\mu_0}\\right)} {\\beta_2}^{2}\\right)} \\sqrt{B + 1} m \\log\\left(\\frac{\\sqrt{B + 1} \\sqrt{m} y - 2 \\, m}{\\sqrt{B + 1} \\sqrt{m} y + 2 \\, m}\\right) - 2 \\, {\\left(4 \\, {\\left(\\sqrt{2} {\\mu_0} {\\beta_1}^{2} - \\sqrt{2} {\\mu_0} {\\beta_2}^{2}\\right)} \\sqrt{B + 1} {\\ell} m \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{B + 1}}\\right) + 2 \\, {\\left(\\sqrt{2} {\\mu_0} {\\beta_1}^{2} - \\sqrt{2} {\\mu_0} {\\beta_2}^{2}\\right)} \\sqrt{B - 1} {\\ell} m \\log\\left(\\frac{\\sqrt{2} \\sqrt{B - 1} {\\ell} y - B + 1}{\\sqrt{2} \\sqrt{B - 1} {\\ell} y + B - 1}\\right) - {\\left({\\left(B^{3} - B\\right)} {\\mu_0} {\\beta_1}^{2} - {\\left(B^{3} - B\\right)} {\\mu_0} {\\beta_2}^{2}\\right)} y\\right)} \\sqrt{m}\\right)} \\sqrt{-{\\mu_0}^{2} + 1}}{4 \\, m}$$" ], "text/plain": [ "1/4*(16*B*C_1*m^(3/2) - (2*(2*(B^2 - 1)*Mu0*beta1 + (4*Mu0*P^2 - (B^2 - 1)*Mu0)*beta1^2 - 2*(B^2 - 1)*Mu0*beta2 - (4*Mu0*Q^2 - (B^2 - 1)*Mu0)*beta2^2)*sqrt(B - 1)*m*arctan(1/2*sqrt(B - 1)*y/sqrt(m)) - (2*(B^2 - 1)*Mu0*beta1 + (4*Mu0*P^2 - (B^2 - 1)*Mu0)*beta1^2 - 2*(B^2 - 1)*Mu0*beta2 - (4*Mu0*Q^2 - (B^2 - 1)*Mu0)*beta2^2)*sqrt(B + 1)*m*log((sqrt(B + 1)*sqrt(m)*y - 2*m)/(sqrt(B + 1)*sqrt(m)*y + 2*m)) - 2*(4*(sqrt(2)*Mu0*beta1^2 - sqrt(2)*Mu0*beta2^2)*sqrt(B + 1)*l*m*arctan(sqrt(2)*l*y/sqrt(B + 1)) + 2*(sqrt(2)*Mu0*beta1^2 - sqrt(2)*Mu0*beta2^2)*sqrt(B - 1)*l*m*log((sqrt(2)*sqrt(B - 1)*l*y - B + 1)/(sqrt(2)*sqrt(B - 1)*l*y + B - 1)) - ((B^3 - B)*Mu0*beta1^2 - (B^3 - B)*Mu0*beta2^2)*y)*sqrt(m))*sqrt(-Mu0^2 + 1))/m" ] }, "execution_count": 114, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pi_theta_y_a2sT0.numerator().subs({l^2: (B^2 - 1)/(8*m)}).simplify_full()" ] }, { "cell_type": "code", "execution_count": 115, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0} \\sqrt{m}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0} \\sqrt{m}$$" ], "text/plain": [ "4*sqrt(8*l^2*m + 1)*sqrt(-Mu0^2 + 1)*Mu0*sqrt(m)" ] }, "execution_count": 115, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pi_theta_y_a2sT0.denominator()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The quantity \n", "$$\\frac{\\pi_\\theta^y}{(a^2/2) \\sin 2\\Theta_0} + (\\beta_1^2 - \\beta_2^2)\\ell^2 y - \\frac{C_1}{\\sin\\Theta_0\\cos\\Theta_0}$$" ] }, { "cell_type": "code", "execution_count": 116, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(2 \\, {\\left(\\sqrt{2} {\\beta_1}^{2} - \\sqrt{2} {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}\\right) + {\\left({\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\log\\left(\\frac{\\sqrt{m} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - 2 \\, m}{\\sqrt{m} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} + 2 \\, m}\\right)\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} + {\\left({\\left(\\sqrt{2} {\\beta_1}^{2} - \\sqrt{2} {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\log\\left(\\frac{\\sqrt{2} {\\ell} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}{\\sqrt{2} {\\ell} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}\\right) - 2 \\, {\\left({\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\arctan\\left(\\frac{y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{2 \\, \\sqrt{m}}\\right)\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{m}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(2 \\, {\\left(\\sqrt{2} {\\beta_1}^{2} - \\sqrt{2} {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\arctan\\left(\\frac{\\sqrt{2} {\\ell} y}{\\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}}\\right) + {\\left({\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\log\\left(\\frac{\\sqrt{m} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - 2 \\, m}{\\sqrt{m} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} + 2 \\, m}\\right)\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} + {\\left({\\left(\\sqrt{2} {\\beta_1}^{2} - \\sqrt{2} {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\log\\left(\\frac{\\sqrt{2} {\\ell} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} - \\sqrt{8 \\, {\\ell}^{2} m + 1} + 1}{\\sqrt{2} {\\ell} y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1} + \\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}\\right) - 2 \\, {\\left({\\mathcal{P}'}^{2} {\\beta_1}^{2} - {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left({\\beta_1}^{2} - {\\beta_2}^{2} - 2 \\, {\\beta_1} + 2 \\, {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\arctan\\left(\\frac{y \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{2 \\, \\sqrt{m}}\\right)\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{m}}$$" ], "text/plain": [ "1/4*((2*(sqrt(2)*beta1^2 - sqrt(2)*beta2^2)*l*sqrt(m)*arctan(sqrt(2)*l*y/sqrt(sqrt(8*l^2*m + 1) + 1)) + (P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*beta1 + 2*beta2)*l^2*m)*log((sqrt(m)*y*sqrt(sqrt(8*l^2*m + 1) + 1) - 2*m)/(sqrt(m)*y*sqrt(sqrt(8*l^2*m + 1) + 1) + 2*m)))*sqrt(sqrt(8*l^2*m + 1) + 1) + ((sqrt(2)*beta1^2 - sqrt(2)*beta2^2)*l*sqrt(m)*log((sqrt(2)*l*y*sqrt(sqrt(8*l^2*m + 1) - 1) - sqrt(8*l^2*m + 1) + 1)/(sqrt(2)*l*y*sqrt(sqrt(8*l^2*m + 1) - 1) + sqrt(8*l^2*m + 1) - 1)) - 2*(P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2 - 2*beta1 + 2*beta2)*l^2*m)*arctan(1/2*y*sqrt(sqrt(8*l^2*m + 1) - 1)/sqrt(m)))*sqrt(sqrt(8*l^2*m + 1) - 1))/(sqrt(8*l^2*m + 1)*sqrt(m))" ] }, "execution_count": 116, "metadata": { }, "output_type": "execute_result" } ], "source": [ "part1 = - (beta1^2 - beta2^2)*l^2*y + C1/(Mu0*sqrt(1-Mu0^2))\n", "s = (pi_theta_y_a2sT0 - part1).simplify_full()\n", "s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Let us perform an expansion in $1/y$ for $y\\rightarrow +\\infty$:" ] }, { "cell_type": "code", "execution_count": 117, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(\\sqrt{2} \\pi {\\beta_1}^{2} - \\sqrt{2} \\pi {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - {\\left(\\pi {\\mathcal{P}'}^{2} {\\beta_1}^{2} - \\pi {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left(\\pi {\\beta_1}^{2} - \\pi {\\beta_2}^{2} - 2 \\, \\pi {\\beta_1} + 2 \\, \\pi {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{m}} - \\frac{{\\beta_1}^{2} - {\\beta_2}^{2}}{y}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(\\sqrt{2} \\pi {\\beta_1}^{2} - \\sqrt{2} \\pi {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - {\\left(\\pi {\\mathcal{P}'}^{2} {\\beta_1}^{2} - \\pi {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left(\\pi {\\beta_1}^{2} - \\pi {\\beta_2}^{2} - 2 \\, \\pi {\\beta_1} + 2 \\, \\pi {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{m}} - \\frac{{\\beta_1}^{2} - {\\beta_2}^{2}}{y}$$" ], "text/plain": [ "1/4*((sqrt(2)*pi*beta1^2 - sqrt(2)*pi*beta2^2)*l*sqrt(m)*sqrt(sqrt(8*l^2*m + 1) + 1) - (pi*P^2*beta1^2 - pi*Q^2*beta2^2 - 2*(pi*beta1^2 - pi*beta2^2 - 2*pi*beta1 + 2*pi*beta2)*l^2*m)*sqrt(sqrt(8*l^2*m + 1) - 1))/(sqrt(8*l^2*m + 1)*sqrt(m)) - (beta1^2 - beta2^2)/y" ] }, "execution_count": 117, "metadata": { }, "output_type": "execute_result" } ], "source": [ "u = var('u')\n", "assume(u > 0)\n", "s = s.subs({y: 1/u}).simplify_log()\n", "assume(l>0)\n", "s = s.taylor(u, 0, 2)\n", "s = s.subs({u: 1/y})\n", "s" ] }, { "cell_type": "code", "execution_count": 118, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(\\sqrt{2} \\pi {\\beta_1}^{2} - \\sqrt{2} \\pi {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - {\\left(\\pi {\\mathcal{P}'}^{2} {\\beta_1}^{2} - \\pi {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left(\\pi {\\beta_1}^{2} - \\pi {\\beta_2}^{2} - 2 \\, \\pi {\\beta_1} + 2 \\, \\pi {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{m}} - \\frac{{\\beta_1}^{2} - {\\beta_2}^{2}}{y}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(\\sqrt{2} \\pi {\\beta_1}^{2} - \\sqrt{2} \\pi {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - {\\left(\\pi {\\mathcal{P}'}^{2} {\\beta_1}^{2} - \\pi {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left(\\pi {\\beta_1}^{2} - \\pi {\\beta_2}^{2} - 2 \\, \\pi {\\beta_1} + 2 \\, \\pi {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{m}} - \\frac{{\\beta_1}^{2} - {\\beta_2}^{2}}{y}$$" ], "text/plain": [ "1/4*((sqrt(2)*pi*beta1^2 - sqrt(2)*pi*beta2^2)*l*sqrt(m)*sqrt(sqrt(8*l^2*m + 1) + 1) - (pi*P^2*beta1^2 - pi*Q^2*beta2^2 - 2*(pi*beta1^2 - pi*beta2^2 - 2*pi*beta1 + 2*pi*beta2)*l^2*m)*sqrt(sqrt(8*l^2*m + 1) - 1))/(sqrt(8*l^2*m + 1)*sqrt(m)) - (beta1^2 - beta2^2)/y" ] }, "execution_count": 118, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Final result for $\\frac{\\pi_\\theta^y}{(a^2/2) \\sin 2\\Theta_0}$:" ] }, { "cell_type": "code", "execution_count": 119, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y + \\frac{{\\left(\\sqrt{2} \\pi {\\beta_1}^{2} - \\sqrt{2} \\pi {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - {\\left(\\pi {\\mathcal{P}'}^{2} {\\beta_1}^{2} - \\pi {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left(\\pi {\\beta_1}^{2} - \\pi {\\beta_2}^{2} - 2 \\, \\pi {\\beta_1} + 2 \\, \\pi {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{m}} - \\frac{{\\beta_1}^{2} - {\\beta_2}^{2}}{y} + \\frac{C_{1}}{\\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0}}$ " ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left({\\beta_1}^{2} - {\\beta_2}^{2}\\right)} {\\ell}^{2} y + \\frac{{\\left(\\sqrt{2} \\pi {\\beta_1}^{2} - \\sqrt{2} \\pi {\\beta_2}^{2}\\right)} {\\ell} \\sqrt{m} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} + 1} - {\\left(\\pi {\\mathcal{P}'}^{2} {\\beta_1}^{2} - \\pi {\\mathcal{Q}'}^{2} {\\beta_2}^{2} - 2 \\, {\\left(\\pi {\\beta_1}^{2} - \\pi {\\beta_2}^{2} - 2 \\, \\pi {\\beta_1} + 2 \\, \\pi {\\beta_2}\\right)} {\\ell}^{2} m\\right)} \\sqrt{\\sqrt{8 \\, {\\ell}^{2} m + 1} - 1}}{4 \\, \\sqrt{8 \\, {\\ell}^{2} m + 1} \\sqrt{m}} - \\frac{{\\beta_1}^{2} - {\\beta_2}^{2}}{y} + \\frac{C_{1}}{\\sqrt{-{\\mu_0}^{2} + 1} {\\mu_0}}$$" ], "text/plain": [ "-(beta1^2 - beta2^2)*l^2*y + 1/4*((sqrt(2)*pi*beta1^2 - sqrt(2)*pi*beta2^2)*l*sqrt(m)*sqrt(sqrt(8*l^2*m + 1) + 1) - (pi*P^2*beta1^2 - pi*Q^2*beta2^2 - 2*(pi*beta1^2 - pi*beta2^2 - 2*pi*beta1 + 2*pi*beta2)*l^2*m)*sqrt(sqrt(8*l^2*m + 1) - 1))/(sqrt(8*l^2*m + 1)*sqrt(m)) - (beta1^2 - beta2^2)/y + C_1/(sqrt(-Mu0^2 + 1)*Mu0)" ] }, "execution_count": 119, "metadata": { }, "output_type": "execute_result" } ], "source": [ "part1 + s" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The terms in $C_1$, $y$ and $y^{-1}$ agree with Eq. (4.37) of the paper." ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.3", "language": "sagemath", "metadata": { "cocalc": { "description": "Open-source mathematical software system", "priority": 1, "url": "https://www.sagemath.org/" } }, "name": "sage-9.3", "resource_dir": "/ext/jupyter/kernels/sage-9.3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.2" } }, "nbformat": 4, "nbformat_minor": 4 }