Higher-Order Equation of Motion

Our goal is to solve the equation of motion for the high-order acceleration $f$, a given higher-order Lagrangian $L$.

Calculations are done using SageManifolds.

%typeset_mode True

from sage.manifolds.utilities import exterior_derivative as ddef ev(N): return (lambda x: N.contract(x))

Variables

M = Manifold(5,'M')coord.<t, x, v, a, s> = M.chart()

Vectors (partial derivatives)

[Dt,Dx,Dv,Da,Ds] = coord.frame()

Forms

[dt,dx,dv,da,ds] = coord.coframe()

General Lagrangian

L = M.scalar_field(function('L')(*list(coord))); L.display()

$\displaystyle \begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & L\left(t, x, v, a, s\right) \end{array}$

Kinematics

$f$ is an unknown function that is the analogue of acceleration for a higher-order equation of motion.

f = M.scalar_field(function('f')(*list(coord))); f.display()

$\displaystyle \begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & f\left(t, x, v, a, s\right) \end{array}$

N = Dt + v*Dx + a*Dv + s*Da + f*Ds; N.display()

$\displaystyle \frac{\partial}{\partial t } + v \frac{\partial}{\partial x } + a \frac{\partial}{\partial v } + s \frac{\partial}{\partial a } + f\left(t, x, v, a, s\right) \frac{\partial}{\partial s }$

The Equation of Lagrange can be defined with the aid of the following auxillary fields

r=Ds(L); r.display()

$\displaystyle \begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & \frac{\partial\,L}{\partial s} \end{array}$

q=Da(L)-N(r); q.display()

$\displaystyle \begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & -v \frac{\partial^2\,L}{\partial x\partial s} - a \frac{\partial^2\,L}{\partial v\partial s} - s \frac{\partial^2\,L}{\partial a\partial s} - f\left(t, x, v, a, s\right) \frac{\partial^2\,L}{\partial s ^ 2} - \frac{\partial^2\,L}{\partial t\partial s} + \frac{\partial\,L}{\partial a} \end{array}$

p=Dv(L)-N(q); p.display()

$\displaystyle \begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & v^{2} \frac{\partial^3\,L}{\partial x ^ 2\partial s} + a^{2} \frac{\partial^3\,L}{\partial v ^ 2\partial s} + 2 \, a s \frac{\partial^3\,L}{\partial v\partial a\partial s} + 2 \, a f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial v\partial s ^ 2} + s^{2} \frac{\partial^3\,L}{\partial a ^ 2\partial s} + 2 \, s f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial a\partial s ^ 2} + f\left(t, x, v, a, s\right)^{2} \frac{\partial^3\,L}{\partial s ^ 3} + {\left(2 \, a \frac{\partial^3\,L}{\partial x\partial v\partial s} + 2 \, s \frac{\partial^3\,L}{\partial x\partial a\partial s} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial x\partial s ^ 2} + \frac{\partial^2\,L}{\partial s ^ 2} \frac{\partial\,f}{\partial x} + 2 \, \frac{\partial^3\,L}{\partial t\partial x\partial s} - \frac{\partial^2\,L}{\partial x\partial a}\right)} v + 2 \, a \frac{\partial^3\,L}{\partial t\partial v\partial s} + 2 \, s \frac{\partial^3\,L}{\partial t\partial a\partial s} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial t\partial s ^ 2} + a \frac{\partial^2\,L}{\partial x\partial s} - a \frac{\partial^2\,L}{\partial v\partial a} + s \frac{\partial^2\,L}{\partial v\partial s} - s \frac{\partial^2\,L}{\partial a ^ 2} + {\left(a \frac{\partial\,f}{\partial v} + s \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + \frac{\partial\,f}{\partial t}\right)} \frac{\partial^2\,L}{\partial s ^ 2} + \frac{\partial^3\,L}{\partial t ^ 2\partial s} - \frac{\partial^2\,L}{\partial t\partial a} + \frac{\partial\,L}{\partial v} \end{array}$

eq1 = (N(p)-Dx(L)); eq1.display()

$\displaystyle \begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & v^{3} \frac{\partial^4\,L}{\partial x ^ 3\partial s} + a^{3} \frac{\partial^4\,L}{\partial v ^ 3\partial s} + 3 \, a^{2} s \frac{\partial^4\,L}{\partial v ^ 2\partial a\partial s} + 3 \, a^{2} f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial v ^ 2\partial s ^ 2} + 3 \, a s^{2} \frac{\partial^4\,L}{\partial v\partial a ^ 2\partial s} + 6 \, a s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial v\partial a\partial s ^ 2} + 3 \, a f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial v\partial s ^ 3} + s^{3} \frac{\partial^4\,L}{\partial a ^ 3\partial s} + 3 \, s^{2} f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial a ^ 2\partial s ^ 2} + 3 \, s f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial a\partial s ^ 3} + f\left(t, x, v, a, s\right)^{3} \frac{\partial^4\,L}{\partial s ^ 4} + {\left(3 \, a \frac{\partial^4\,L}{\partial x ^ 2\partial v\partial s} + 3 \, s \frac{\partial^4\,L}{\partial x ^ 2\partial a\partial s} + 3 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x ^ 2\partial s ^ 2} + 3 \, \frac{\partial^3\,L}{\partial x\partial s ^ 2} \frac{\partial\,f}{\partial x} + \frac{\partial^2\,L}{\partial s ^ 2} \frac{\partial^2\,f}{\partial x ^ 2} + 3 \, \frac{\partial^4\,L}{\partial t\partial x ^ 2\partial s} - \frac{\partial^3\,L}{\partial x ^ 2\partial a}\right)} v^{2} + 3 \, a^{2} \frac{\partial^4\,L}{\partial t\partial v ^ 2\partial s} + 6 \, a s \frac{\partial^4\,L}{\partial t\partial v\partial a\partial s} + 6 \, a f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial v\partial s ^ 2} + 3 \, s^{2} \frac{\partial^4\,L}{\partial t\partial a ^ 2\partial s} + 6 \, s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial a\partial s ^ 2} + 3 \, f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial t\partial s ^ 3} + 3 \, a^{2} \frac{\partial^3\,L}{\partial x\partial v\partial s} + 3 \, a s \frac{\partial^3\,L}{\partial x\partial a\partial s} + 3 \, a f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial x\partial s ^ 2} - a^{2} \frac{\partial^3\,L}{\partial v ^ 2\partial a} + 3 \, a s \frac{\partial^3\,L}{\partial v ^ 2\partial s} - 2 \, a s \frac{\partial^3\,L}{\partial v\partial a ^ 2} - s^{2} \frac{\partial^3\,L}{\partial a ^ 3} + s f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial a ^ 2\partial s} + {\left(3 \, a^{2} \frac{\partial^4\,L}{\partial x\partial v ^ 2\partial s} + 6 \, a s \frac{\partial^4\,L}{\partial x\partial v\partial a\partial s} + 6 \, a f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x\partial v\partial s ^ 2} + 3 \, s^{2} \frac{\partial^4\,L}{\partial x\partial a ^ 2\partial s} + 6 \, s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x\partial a\partial s ^ 2} + 3 \, f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial x\partial s ^ 3} + 3 \, a \frac{\partial^3\,L}{\partial v\partial s ^ 2} \frac{\partial\,f}{\partial x} + 3 \, s \frac{\partial^3\,L}{\partial a\partial s ^ 2} \frac{\partial\,f}{\partial x} + 3 \, f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial s ^ 3} \frac{\partial\,f}{\partial x} + 6 \, a \frac{\partial^4\,L}{\partial t\partial x\partial v\partial s} + 6 \, s \frac{\partial^4\,L}{\partial t\partial x\partial a\partial s} + 6 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial x\partial s ^ 2} + 3 \, a \frac{\partial^3\,L}{\partial x ^ 2\partial s} - 2 \, a \frac{\partial^3\,L}{\partial x\partial v\partial a} + 3 \, s \frac{\partial^3\,L}{\partial x\partial v\partial s} - 2 \, s \frac{\partial^3\,L}{\partial x\partial a ^ 2} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial x\partial a\partial s} + 3 \, {\left(a \frac{\partial\,f}{\partial v} + s \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial x\partial s ^ 2} + {\left(2 \, a \frac{\partial^2\,f}{\partial x\partial v} + 2 \, s \frac{\partial^2\,f}{\partial x\partial a} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^2\,f}{\partial x\partial s} + \frac{\partial\,f}{\partial x} \frac{\partial\,f}{\partial s} + 2 \, \frac{\partial^2\,f}{\partial t\partial x}\right)} \frac{\partial^2\,L}{\partial s ^ 2} + 3 \, \frac{\partial^3\,L}{\partial t\partial s ^ 2} \frac{\partial\,f}{\partial x} + 3 \, \frac{\partial^4\,L}{\partial t ^ 2\partial x\partial s} - 2 \, \frac{\partial^3\,L}{\partial t\partial x\partial a} + \frac{\partial^2\,L}{\partial x\partial v}\right)} v + 3 \, a \frac{\partial^4\,L}{\partial t ^ 2\partial v\partial s} + 3 \, s \frac{\partial^4\,L}{\partial t ^ 2\partial a\partial s} + 3 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t ^ 2\partial s ^ 2} + 3 \, a \frac{\partial^3\,L}{\partial t\partial x\partial s} - 2 \, a \frac{\partial^3\,L}{\partial t\partial v\partial a} + 3 \, s \frac{\partial^3\,L}{\partial t\partial v\partial s} - 2 \, s \frac{\partial^3\,L}{\partial t\partial a ^ 2} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial t\partial a\partial s} + 3 \, {\left(a \frac{\partial\,f}{\partial v} + s \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial t\partial s ^ 2} - a \frac{\partial^2\,L}{\partial x\partial a} + s \frac{\partial^2\,L}{\partial x\partial s} + a \frac{\partial^2\,L}{\partial v ^ 2} + {\left(3 \, s^{2} + a f\left(t, x, v, a, s\right)\right)} \frac{\partial^3\,L}{\partial v\partial a\partial s} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^2\,L}{\partial v\partial s} + 3 \, {\left(a^{2} \frac{\partial\,f}{\partial v} + a s \frac{\partial\,f}{\partial a} + a f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + s f\left(t, x, v, a, s\right) + a \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial v\partial s ^ 2} - f\left(t, x, v, a, s\right) \frac{\partial^2\,L}{\partial a ^ 2} + {\left(3 \, a s \frac{\partial\,f}{\partial v} + 3 \, s^{2} \frac{\partial\,f}{\partial a} + 3 \, s f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + 2 \, f\left(t, x, v, a, s\right)^{2} + 3 \, s \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial a\partial s ^ 2} + {\left(a^{2} \frac{\partial^2\,f}{\partial v ^ 2} + 2 \, a s \frac{\partial^2\,f}{\partial v\partial a} + 2 \, a f\left(t, x, v, a, s\right) \frac{\partial^2\,f}{\partial v\partial s} + s^{2} \frac{\partial^2\,f}{\partial a ^ 2} + 2 \, s f\left(t, x, v, a, s\right) \frac{\partial^2\,f}{\partial a\partial s} + s \frac{\partial\,f}{\partial a} \frac{\partial\,f}{\partial s} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s}^{2} + f\left(t, x, v, a, s\right)^{2} \frac{\partial^2\,f}{\partial s ^ 2} + 2 \, a \frac{\partial^2\,f}{\partial t\partial v} + 2 \, s \frac{\partial^2\,f}{\partial t\partial a} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^2\,f}{\partial t\partial s} + a \frac{\partial\,f}{\partial x} + {\left(a \frac{\partial\,f}{\partial s} + s\right)} \frac{\partial\,f}{\partial v} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial a} + \frac{\partial\,f}{\partial t} \frac{\partial\,f}{\partial s} + \frac{\partial^2\,f}{\partial t ^ 2}\right)} \frac{\partial^2\,L}{\partial s ^ 2} + 3 \, {\left(a f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial v} + s f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right)^{2} \frac{\partial\,f}{\partial s} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial s ^ 3} + \frac{\partial^4\,L}{\partial t ^ 3\partial s} - \frac{\partial^3\,L}{\partial t ^ 2\partial a} + \frac{\partial^2\,L}{\partial t\partial v} - \frac{\partial\,L}{\partial x} \end{array}$

Number of terms in this expression:

len(eq1.expr().expand())

$\displaystyle 116$

We want to solve this equation (eq1 = 0) for $f$ in terms of $L$.

The equation can also be written in the following form:

eq2 = N(Dv(L)) - N(N(Da(L))) + N(N(N(Ds(L)))) - Dx(L)

eq1 == eq2

$\displaystyle \mathrm{True}$

But we would prefer to write it as a polynomial in $N(N(f))$, $N(f)$, and powers of $f$. If we write $f$ instead of $N(Ds(L))$, $N(Da(L))$ and $N(Dv(L))$ above then we get new coefficients and terms in powers of $f$.

N(N(f)).display()

$\displaystyle \begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & v^{2} \frac{\partial^2\,f}{\partial x ^ 2} + a^{2} \frac{\partial^2\,f}{\partial v ^ 2} + 2 \, a s \frac{\partial^2\,f}{\partial v\partial a} + 2 \, a f\left(t, x, v, a, s\right) \frac{\partial^2\,f}{\partial v\partial s} + s^{2} \frac{\partial^2\,f}{\partial a ^ 2} + 2 \, s f\left(t, x, v, a, s\right) \frac{\partial^2\,f}{\partial a\partial s} + s \frac{\partial\,f}{\partial a} \frac{\partial\,f}{\partial s} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s}^{2} + f\left(t, x, v, a, s\right)^{2} \frac{\partial^2\,f}{\partial s ^ 2} + {\left(2 \, a \frac{\partial^2\,f}{\partial x\partial v} + 2 \, s \frac{\partial^2\,f}{\partial x\partial a} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^2\,f}{\partial x\partial s} + \frac{\partial\,f}{\partial x} \frac{\partial\,f}{\partial s} + 2 \, \frac{\partial^2\,f}{\partial t\partial x}\right)} v + 2 \, a \frac{\partial^2\,f}{\partial t\partial v} + 2 \, s \frac{\partial^2\,f}{\partial t\partial a} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^2\,f}{\partial t\partial s} + a \frac{\partial\,f}{\partial x} + {\left(a \frac{\partial\,f}{\partial s} + s\right)} \frac{\partial\,f}{\partial v} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial a} + \frac{\partial\,f}{\partial t} \frac{\partial\,f}{\partial s} + \frac{\partial^2\,f}{\partial t ^ 2} \end{array}$

Terms 2$^\textit{nd}$ order in $f$ arise from the Liebniz rule applied to the differential operator $N^2$ on the term $Ds\ f$ in $N(Ds(L))$. For example

t2 = eq1.expr().coefficient(diff(f.expr(),t,t))*diff(f.expr(),t,t); M.scalar_field(t2).display()

$\displaystyle \begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & \frac{\partial^2\,L}{\partial s ^ 2} \frac{\partial^2\,f}{\partial t ^ 2} \end{array}$

The necessary coefficient can be read from the above.

c2 = (Ds(Ds(L))*N(N(f))); c2.display()

After removing the term $c2$ above, the remaining terms are at most first-order in $f$.

eq1a = eq1-c2; eq1a.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & v^{3} \frac{\partial^4\,L}{\partial x ^ 3\partial s} + a^{3} \frac{\partial^4\,L}{\partial v ^ 3\partial s} + 3 \, a^{2} s \frac{\partial^4\,L}{\partial v ^ 2\partial a\partial s} + 3 \, a^{2} f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial v ^ 2\partial s ^ 2} + 3 \, a s^{2} \frac{\partial^4\,L}{\partial v\partial a ^ 2\partial s} + 6 \, a s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial v\partial a\partial s ^ 2} + 3 \, a f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial v\partial s ^ 3} + s^{3} \frac{\partial^4\,L}{\partial a ^ 3\partial s} + 3 \, s^{2} f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial a ^ 2\partial s ^ 2} + 3 \, s f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial a\partial s ^ 3} + f\left(t, x, v, a, s\right)^{3} \frac{\partial^4\,L}{\partial s ^ 4} + {\left(3 \, a \frac{\partial^4\,L}{\partial x ^ 2\partial v\partial s} + 3 \, s \frac{\partial^4\,L}{\partial x ^ 2\partial a\partial s} + 3 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x ^ 2\partial s ^ 2} + 3 \, \frac{\partial^3\,L}{\partial x\partial s ^ 2} \frac{\partial\,f}{\partial x} + 3 \, \frac{\partial^4\,L}{\partial t\partial x ^ 2\partial s} - \frac{\partial^3\,L}{\partial x ^ 2\partial a}\right)} v^{2} + 3 \, a^{2} \frac{\partial^4\,L}{\partial t\partial v ^ 2\partial s} + 6 \, a s \frac{\partial^4\,L}{\partial t\partial v\partial a\partial s} + 6 \, a f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial v\partial s ^ 2} + 3 \, s^{2} \frac{\partial^4\,L}{\partial t\partial a ^ 2\partial s} + 6 \, s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial a\partial s ^ 2} + 3 \, f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial t\partial s ^ 3} + 3 \, a^{2} \frac{\partial^3\,L}{\partial x\partial v\partial s} + 3 \, a s \frac{\partial^3\,L}{\partial x\partial a\partial s} + 3 \, a f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial x\partial s ^ 2} - a^{2} \frac{\partial^3\,L}{\partial v ^ 2\partial a} + 3 \, a s \frac{\partial^3\,L}{\partial v ^ 2\partial s} - 2 \, a s \frac{\partial^3\,L}{\partial v\partial a ^ 2} - s^{2} \frac{\partial^3\,L}{\partial a ^ 3} + s f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial a ^ 2\partial s} + {\left(3 \, a^{2} \frac{\partial^4\,L}{\partial x\partial v ^ 2\partial s} + 6 \, a s \frac{\partial^4\,L}{\partial x\partial v\partial a\partial s} + 6 \, a f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x\partial v\partial s ^ 2} + 3 \, s^{2} \frac{\partial^4\,L}{\partial x\partial a ^ 2\partial s} + 6 \, s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x\partial a\partial s ^ 2} + 3 \, f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial x\partial s ^ 3} + 3 \, a \frac{\partial^3\,L}{\partial v\partial s ^ 2} \frac{\partial\,f}{\partial x} + 3 \, s \frac{\partial^3\,L}{\partial a\partial s ^ 2} \frac{\partial\,f}{\partial x} + 3 \, f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial s ^ 3} \frac{\partial\,f}{\partial x} + 6 \, a \frac{\partial^4\,L}{\partial t\partial x\partial v\partial s} + 6 \, s \frac{\partial^4\,L}{\partial t\partial x\partial a\partial s} + 6 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial x\partial s ^ 2} + 3 \, a \frac{\partial^3\,L}{\partial x ^ 2\partial s} - 2 \, a \frac{\partial^3\,L}{\partial x\partial v\partial a} + 3 \, s \frac{\partial^3\,L}{\partial x\partial v\partial s} - 2 \, s \frac{\partial^3\,L}{\partial x\partial a ^ 2} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial x\partial a\partial s} + 3 \, {\left(a \frac{\partial\,f}{\partial v} + s \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial x\partial s ^ 2} + 3 \, \frac{\partial^3\,L}{\partial t\partial s ^ 2} \frac{\partial\,f}{\partial x} + 3 \, \frac{\partial^4\,L}{\partial t ^ 2\partial x\partial s} - 2 \, \frac{\partial^3\,L}{\partial t\partial x\partial a} + \frac{\partial^2\,L}{\partial x\partial v}\right)} v + 3 \, a \frac{\partial^4\,L}{\partial t ^ 2\partial v\partial s} + 3 \, s \frac{\partial^4\,L}{\partial t ^ 2\partial a\partial s} + 3 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t ^ 2\partial s ^ 2} + 3 \, a \frac{\partial^3\,L}{\partial t\partial x\partial s} - 2 \, a \frac{\partial^3\,L}{\partial t\partial v\partial a} + 3 \, s \frac{\partial^3\,L}{\partial t\partial v\partial s} - 2 \, s \frac{\partial^3\,L}{\partial t\partial a ^ 2} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial t\partial a\partial s} + 3 \, {\left(a \frac{\partial\,f}{\partial v} + s \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial t\partial s ^ 2} - a \frac{\partial^2\,L}{\partial x\partial a} + s \frac{\partial^2\,L}{\partial x\partial s} + a \frac{\partial^2\,L}{\partial v ^ 2} + {\left(3 \, s^{2} + a f\left(t, x, v, a, s\right)\right)} \frac{\partial^3\,L}{\partial v\partial a\partial s} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^2\,L}{\partial v\partial s} + 3 \, {\left(a^{2} \frac{\partial\,f}{\partial v} + a s \frac{\partial\,f}{\partial a} + a f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + s f\left(t, x, v, a, s\right) + a \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial v\partial s ^ 2} - f\left(t, x, v, a, s\right) \frac{\partial^2\,L}{\partial a ^ 2} + {\left(3 \, a s \frac{\partial\,f}{\partial v} + 3 \, s^{2} \frac{\partial\,f}{\partial a} + 3 \, s f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + 2 \, f\left(t, x, v, a, s\right)^{2} + 3 \, s \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial a\partial s ^ 2} + 3 \, {\left(a f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial v} + s f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right)^{2} \frac{\partial\,f}{\partial s} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial s ^ 3} + \frac{\partial^4\,L}{\partial t ^ 3\partial s} - \frac{\partial^3\,L}{\partial t ^ 2\partial a} + \frac{\partial^2\,L}{\partial t\partial v} - \frac{\partial\,L}{\partial x} \end{array}$

Similarly the first-order terms arise from $N$ applied to the term $Ds\ f$ in $N(Da(L))$.

N(f).display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & v \frac{\partial\,f}{\partial x} + a \frac{\partial\,f}{\partial v} + s \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + \frac{\partial\,f}{\partial t} \end{array}$

For example:

t1 = eq1a.expr().coefficient(diff(f.expr(),t))*diff(f.expr(),t); M.scalar_field(t1).display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & 3 \, {\left(v \frac{\partial^3\,L}{\partial x\partial s ^ 2} + a \frac{\partial^3\,L}{\partial v\partial s ^ 2} + s \frac{\partial^3\,L}{\partial a\partial s ^ 2} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial s ^ 3} + \frac{\partial^3\,L}{\partial t\partial s ^ 2}\right)} \frac{\partial\,f}{\partial t} \end{array}$

x1 = eq1a.expr().coefficient(diff(f.expr(),x))*diff(f.expr(),x); M.scalar_field(x1).display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & 3 \, {\left(v^{2} \frac{\partial^3\,L}{\partial x\partial s ^ 2} + {\left(a \frac{\partial^3\,L}{\partial v\partial s ^ 2} + s \frac{\partial^3\,L}{\partial a\partial s ^ 2} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial s ^ 3} + \frac{\partial^3\,L}{\partial t\partial s ^ 2}\right)} v\right)} \frac{\partial\,f}{\partial x} \end{array}$

v1 = eq1a.expr().coefficient(diff(f.expr(),v))*diff(f.expr(),v); M.scalar_field(v1).display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & 3 \, {\left(a v \frac{\partial^3\,L}{\partial x\partial s ^ 2} + a^{2} \frac{\partial^3\,L}{\partial v\partial s ^ 2} + a s \frac{\partial^3\,L}{\partial a\partial s ^ 2} + a f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial s ^ 3} + a \frac{\partial^3\,L}{\partial t\partial s ^ 2}\right)} \frac{\partial\,f}{\partial v} \end{array}$

a1 = eq1a.expr().coefficient(diff(f.expr(),a))*diff(f.expr(),a); M.scalar_field(a1).display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & 3 \, {\left(s v \frac{\partial^3\,L}{\partial x\partial s ^ 2} + a s \frac{\partial^3\,L}{\partial v\partial s ^ 2} + s^{2} \frac{\partial^3\,L}{\partial a\partial s ^ 2} + s f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial s ^ 3} + s \frac{\partial^3\,L}{\partial t\partial s ^ 2}\right)} \frac{\partial\,f}{\partial a} \end{array}$

s1 = eq1a.expr().coefficient(diff(f.expr(),s))*diff(f.expr(),s); M.scalar_field(s1).display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & 3 \, {\left(v f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial x\partial s ^ 2} + a f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial v\partial s ^ 2} + s f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial a\partial s ^ 2} + f\left(t, x, v, a, s\right)^{2} \frac{\partial^3\,L}{\partial s ^ 3} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial t\partial s ^ 2}\right)} \frac{\partial\,f}{\partial s} \end{array}$

Liebniz rule applies $N$ to these coefficients.

c1 = 3*N(Ds(Ds(L)))*N(f); c1.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & 3 \, v^{2} \frac{\partial^3\,L}{\partial x\partial s ^ 2} \frac{\partial\,f}{\partial x} + 3 \, {\left(a \frac{\partial^3\,L}{\partial v\partial s ^ 2} \frac{\partial\,f}{\partial x} + s \frac{\partial^3\,L}{\partial a\partial s ^ 2} \frac{\partial\,f}{\partial x} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial s ^ 3} \frac{\partial\,f}{\partial x} + {\left(a \frac{\partial\,f}{\partial v} + s \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial x\partial s ^ 2} + \frac{\partial^3\,L}{\partial t\partial s ^ 2} \frac{\partial\,f}{\partial x}\right)} v + 3 \, {\left(a \frac{\partial\,f}{\partial v} + s \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial t\partial s ^ 2} + 3 \, {\left(a^{2} \frac{\partial\,f}{\partial v} + a s \frac{\partial\,f}{\partial a} + a f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + a \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial v\partial s ^ 2} + 3 \, {\left(a s \frac{\partial\,f}{\partial v} + s^{2} \frac{\partial\,f}{\partial a} + s f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + s \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial a\partial s ^ 2} + 3 \, {\left(a f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial v} + s f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right)^{2} \frac{\partial\,f}{\partial s} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial s ^ 3} \end{array}$

bool(c1.expr()==t1+x1+v1+a1+s1)

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

Removing the 1$^\textit{st}$ order terms:

eq1b = eq1a-c1; eq1b.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & v^{3} \frac{\partial^4\,L}{\partial x ^ 3\partial s} + a^{3} \frac{\partial^4\,L}{\partial v ^ 3\partial s} + 3 \, a^{2} s \frac{\partial^4\,L}{\partial v ^ 2\partial a\partial s} + 3 \, a^{2} f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial v ^ 2\partial s ^ 2} + 3 \, a s^{2} \frac{\partial^4\,L}{\partial v\partial a ^ 2\partial s} + 6 \, a s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial v\partial a\partial s ^ 2} + 3 \, a f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial v\partial s ^ 3} + s^{3} \frac{\partial^4\,L}{\partial a ^ 3\partial s} + 3 \, s^{2} f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial a ^ 2\partial s ^ 2} + 3 \, s f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial a\partial s ^ 3} + f\left(t, x, v, a, s\right)^{3} \frac{\partial^4\,L}{\partial s ^ 4} + {\left(3 \, a \frac{\partial^4\,L}{\partial x ^ 2\partial v\partial s} + 3 \, s \frac{\partial^4\,L}{\partial x ^ 2\partial a\partial s} + 3 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x ^ 2\partial s ^ 2} + 3 \, \frac{\partial^4\,L}{\partial t\partial x ^ 2\partial s} - \frac{\partial^3\,L}{\partial x ^ 2\partial a}\right)} v^{2} + 3 \, a^{2} \frac{\partial^4\,L}{\partial t\partial v ^ 2\partial s} + 6 \, a s \frac{\partial^4\,L}{\partial t\partial v\partial a\partial s} + 6 \, a f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial v\partial s ^ 2} + 3 \, s^{2} \frac{\partial^4\,L}{\partial t\partial a ^ 2\partial s} + 6 \, s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial a\partial s ^ 2} + 3 \, f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial t\partial s ^ 3} + 3 \, a^{2} \frac{\partial^3\,L}{\partial x\partial v\partial s} + 3 \, a s \frac{\partial^3\,L}{\partial x\partial a\partial s} + 3 \, a f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial x\partial s ^ 2} - a^{2} \frac{\partial^3\,L}{\partial v ^ 2\partial a} + 3 \, a s \frac{\partial^3\,L}{\partial v ^ 2\partial s} - 2 \, a s \frac{\partial^3\,L}{\partial v\partial a ^ 2} + 3 \, s f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial v\partial s ^ 2} - s^{2} \frac{\partial^3\,L}{\partial a ^ 3} + s f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial a ^ 2\partial s} + 2 \, f\left(t, x, v, a, s\right)^{2} \frac{\partial^3\,L}{\partial a\partial s ^ 2} + {\left(3 \, a^{2} \frac{\partial^4\,L}{\partial x\partial v ^ 2\partial s} + 6 \, a s \frac{\partial^4\,L}{\partial x\partial v\partial a\partial s} + 6 \, a f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x\partial v\partial s ^ 2} + 3 \, s^{2} \frac{\partial^4\,L}{\partial x\partial a ^ 2\partial s} + 6 \, s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x\partial a\partial s ^ 2} + 3 \, f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial x\partial s ^ 3} + 6 \, a \frac{\partial^4\,L}{\partial t\partial x\partial v\partial s} + 6 \, s \frac{\partial^4\,L}{\partial t\partial x\partial a\partial s} + 6 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial x\partial s ^ 2} + 3 \, a \frac{\partial^3\,L}{\partial x ^ 2\partial s} - 2 \, a \frac{\partial^3\,L}{\partial x\partial v\partial a} + 3 \, s \frac{\partial^3\,L}{\partial x\partial v\partial s} - 2 \, s \frac{\partial^3\,L}{\partial x\partial a ^ 2} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial x\partial a\partial s} + 3 \, \frac{\partial^4\,L}{\partial t ^ 2\partial x\partial s} - 2 \, \frac{\partial^3\,L}{\partial t\partial x\partial a} + \frac{\partial^2\,L}{\partial x\partial v}\right)} v + 3 \, a \frac{\partial^4\,L}{\partial t ^ 2\partial v\partial s} + 3 \, s \frac{\partial^4\,L}{\partial t ^ 2\partial a\partial s} + 3 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t ^ 2\partial s ^ 2} + 3 \, a \frac{\partial^3\,L}{\partial t\partial x\partial s} - 2 \, a \frac{\partial^3\,L}{\partial t\partial v\partial a} + 3 \, s \frac{\partial^3\,L}{\partial t\partial v\partial s} - 2 \, s \frac{\partial^3\,L}{\partial t\partial a ^ 2} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial t\partial a\partial s} - a \frac{\partial^2\,L}{\partial x\partial a} + s \frac{\partial^2\,L}{\partial x\partial s} + a \frac{\partial^2\,L}{\partial v ^ 2} + {\left(3 \, s^{2} + a f\left(t, x, v, a, s\right)\right)} \frac{\partial^3\,L}{\partial v\partial a\partial s} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^2\,L}{\partial v\partial s} - f\left(t, x, v, a, s\right) \frac{\partial^2\,L}{\partial a ^ 2} + \frac{\partial^4\,L}{\partial t ^ 3\partial s} - \frac{\partial^3\,L}{\partial t ^ 2\partial a} + \frac{\partial^2\,L}{\partial t\partial v} - \frac{\partial\,L}{\partial x} \end{array}$

Only terms algebraic in $f$ remain. The number of terms is:

len(eq1b.expr().expand())

$\newcommand{\Bold}[1]{\mathbf{#1}}68$

The term of highest degree comes from $N^3$ applied to $Ds(L)$:

t0 = eq1b.expr().coefficient(f.expr()^3)*f.expr()^3; M.scalar_field(t0.expand()).display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & f\left(t, x, v, a, s\right)^{3} \frac{\partial^4\,L}{\partial s ^ 4} \end{array}$

cf3 = Ds(Ds(Ds(Ds(L))))*f^3; cf3.display()

Removing the 3$^{rd}$ degree term:

eq1c = eq1b-cf3; eq1c.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & v^{3} \frac{\partial^4\,L}{\partial x ^ 3\partial s} + a^{3} \frac{\partial^4\,L}{\partial v ^ 3\partial s} + 3 \, a^{2} s \frac{\partial^4\,L}{\partial v ^ 2\partial a\partial s} + 3 \, a^{2} f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial v ^ 2\partial s ^ 2} + 3 \, a s^{2} \frac{\partial^4\,L}{\partial v\partial a ^ 2\partial s} + 6 \, a s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial v\partial a\partial s ^ 2} + 3 \, a f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial v\partial s ^ 3} + s^{3} \frac{\partial^4\,L}{\partial a ^ 3\partial s} + 3 \, s^{2} f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial a ^ 2\partial s ^ 2} + 3 \, s f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial a\partial s ^ 3} + {\left(3 \, a \frac{\partial^4\,L}{\partial x ^ 2\partial v\partial s} + 3 \, s \frac{\partial^4\,L}{\partial x ^ 2\partial a\partial s} + 3 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x ^ 2\partial s ^ 2} + 3 \, \frac{\partial^4\,L}{\partial t\partial x ^ 2\partial s} - \frac{\partial^3\,L}{\partial x ^ 2\partial a}\right)} v^{2} + 3 \, a^{2} \frac{\partial^4\,L}{\partial t\partial v ^ 2\partial s} + 6 \, a s \frac{\partial^4\,L}{\partial t\partial v\partial a\partial s} + 6 \, a f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial v\partial s ^ 2} + 3 \, s^{2} \frac{\partial^4\,L}{\partial t\partial a ^ 2\partial s} + 6 \, s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial a\partial s ^ 2} + 3 \, f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial t\partial s ^ 3} + 3 \, a^{2} \frac{\partial^3\,L}{\partial x\partial v\partial s} + 3 \, a s \frac{\partial^3\,L}{\partial x\partial a\partial s} + 3 \, a f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial x\partial s ^ 2} - a^{2} \frac{\partial^3\,L}{\partial v ^ 2\partial a} + 3 \, a s \frac{\partial^3\,L}{\partial v ^ 2\partial s} - 2 \, a s \frac{\partial^3\,L}{\partial v\partial a ^ 2} + 3 \, s f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial v\partial s ^ 2} - s^{2} \frac{\partial^3\,L}{\partial a ^ 3} + s f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial a ^ 2\partial s} + 2 \, f\left(t, x, v, a, s\right)^{2} \frac{\partial^3\,L}{\partial a\partial s ^ 2} + {\left(3 \, a^{2} \frac{\partial^4\,L}{\partial x\partial v ^ 2\partial s} + 6 \, a s \frac{\partial^4\,L}{\partial x\partial v\partial a\partial s} + 6 \, a f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x\partial v\partial s ^ 2} + 3 \, s^{2} \frac{\partial^4\,L}{\partial x\partial a ^ 2\partial s} + 6 \, s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x\partial a\partial s ^ 2} + 3 \, f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial x\partial s ^ 3} + 6 \, a \frac{\partial^4\,L}{\partial t\partial x\partial v\partial s} + 6 \, s \frac{\partial^4\,L}{\partial t\partial x\partial a\partial s} + 6 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial x\partial s ^ 2} + 3 \, a \frac{\partial^3\,L}{\partial x ^ 2\partial s} - 2 \, a \frac{\partial^3\,L}{\partial x\partial v\partial a} + 3 \, s \frac{\partial^3\,L}{\partial x\partial v\partial s} - 2 \, s \frac{\partial^3\,L}{\partial x\partial a ^ 2} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial x\partial a\partial s} + 3 \, \frac{\partial^4\,L}{\partial t ^ 2\partial x\partial s} - 2 \, \frac{\partial^3\,L}{\partial t\partial x\partial a} + \frac{\partial^2\,L}{\partial x\partial v}\right)} v + 3 \, a \frac{\partial^4\,L}{\partial t ^ 2\partial v\partial s} + 3 \, s \frac{\partial^4\,L}{\partial t ^ 2\partial a\partial s} + 3 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t ^ 2\partial s ^ 2} + 3 \, a \frac{\partial^3\,L}{\partial t\partial x\partial s} - 2 \, a \frac{\partial^3\,L}{\partial t\partial v\partial a} + 3 \, s \frac{\partial^3\,L}{\partial t\partial v\partial s} - 2 \, s \frac{\partial^3\,L}{\partial t\partial a ^ 2} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial t\partial a\partial s} - a \frac{\partial^2\,L}{\partial x\partial a} + s \frac{\partial^2\,L}{\partial x\partial s} + a \frac{\partial^2\,L}{\partial v ^ 2} + {\left(3 \, s^{2} + a f\left(t, x, v, a, s\right)\right)} \frac{\partial^3\,L}{\partial v\partial a\partial s} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^2\,L}{\partial v\partial s} - f\left(t, x, v, a, s\right) \frac{\partial^2\,L}{\partial a ^ 2} + \frac{\partial^4\,L}{\partial t ^ 3\partial s} - \frac{\partial^3\,L}{\partial t ^ 2\partial a} + \frac{\partial^2\,L}{\partial t\partial v} - \frac{\partial\,L}{\partial x} \end{array}$

leaves terms of at most 2$^{nd}$ degree. These 2$^{nd}$ degree terms come for $N^2$ applied to $Da(L)$,

t0 = eq1c.expr().coefficient(f.expr()^2)*f.expr()^2; M.scalar_field(t0.expand()).display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & 3 \, v f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial x\partial s ^ 3} + 3 \, a f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial v\partial s ^ 3} + 3 \, s f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial a\partial s ^ 3} + 3 \, f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial t\partial s ^ 3} + 2 \, f\left(t, x, v, a, s\right)^{2} \frac{\partial^3\,L}{\partial a\partial s ^ 2} \end{array}$

cf2 = N(Ds(Da(L)))*f; cf2.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & v f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial x\partial a\partial s} + a f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial v\partial a\partial s} + s f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial a ^ 2\partial s} + f\left(t, x, v, a, s\right)^{2} \frac{\partial^3\,L}{\partial a\partial s ^ 2} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial t\partial a\partial s} \end{array}$

cf2 = Ds(Ds(Da(L)))*f^2; cf2.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & f\left(t, x, v, a, s\right)^{2} \frac{\partial^3\,L}{\partial a\partial s ^ 2} \end{array}$

We cannot solve this for f algebraically since f appears as derivatives.

Lagrangian linear in s.

ll=list(coord);ll.remove(s);ll

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[t, x, v, a\right]$

Ll = M.scalar_field(function('L0')(*ll)) + s * M.scalar_field(function('L1')(*ll)); Ll.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & s L_{1}\left(t, x, v, a\right) + L_{0}\left(t, x, v, a\right) \end{array}$

solve(eq1.expr().substitute_function(L.expr().operator(),Ll.expr().function(*list(coord))),f.expr())[0]

$\newcommand{\Bold}[1]{\mathbf{#1}}f\left(t, x, v, a, s\right) = -\frac{v^{3} \frac{\partial^{3}}{(\partial x)^{3}}L_{1}\left(t, x, v, a\right) + a^{3} \frac{\partial^{3}}{(\partial v)^{3}}L_{1}\left(t, x, v, a\right) + 2 \, a^{2} s \frac{\partial^{3}}{(\partial v)^{2}\partial a}L_{1}\left(t, x, v, a\right) + a s^{2} \frac{\partial^{3}}{\partial v(\partial a)^{2}}L_{1}\left(t, x, v, a\right) + {\left(3 \, a \frac{\partial^{3}}{(\partial x)^{2}\partial v}L_{1}\left(t, x, v, a\right) + 2 \, s \frac{\partial^{3}}{(\partial x)^{2}\partial a}L_{1}\left(t, x, v, a\right) - \frac{\partial^{3}}{(\partial x)^{2}\partial a}L_{0}\left(t, x, v, a\right) + 3 \, \frac{\partial^{3}}{\partial t(\partial x)^{2}}L_{1}\left(t, x, v, a\right)\right)} v^{2} - a^{2} \frac{\partial^{3}}{(\partial v)^{2}\partial a}L_{0}\left(t, x, v, a\right) - 2 \, a s \frac{\partial^{3}}{\partial v(\partial a)^{2}}L_{0}\left(t, x, v, a\right) - s^{2} \frac{\partial^{3}}{(\partial a)^{3}}L_{0}\left(t, x, v, a\right) + 3 \, a^{2} \frac{\partial^{3}}{\partial t(\partial v)^{2}}L_{1}\left(t, x, v, a\right) + 4 \, a s \frac{\partial^{3}}{\partial t\partial v\partial a}L_{1}\left(t, x, v, a\right) + s^{2} \frac{\partial^{3}}{\partial t(\partial a)^{2}}L_{1}\left(t, x, v, a\right) + 3 \, a^{2} \frac{\partial^{2}}{\partial x\partial v}L_{1}\left(t, x, v, a\right) + 2 \, a s \frac{\partial^{2}}{\partial x\partial a}L_{1}\left(t, x, v, a\right) + 4 \, a s \frac{\partial^{2}}{(\partial v)^{2}}L_{1}\left(t, x, v, a\right) + 3 \, s^{2} \frac{\partial^{2}}{\partial v\partial a}L_{1}\left(t, x, v, a\right) + {\left(3 \, a^{2} \frac{\partial^{3}}{\partial x(\partial v)^{2}}L_{1}\left(t, x, v, a\right) + 4 \, a s \frac{\partial^{3}}{\partial x\partial v\partial a}L_{1}\left(t, x, v, a\right) + s^{2} \frac{\partial^{3}}{\partial x(\partial a)^{2}}L_{1}\left(t, x, v, a\right) - 2 \, a \frac{\partial^{3}}{\partial x\partial v\partial a}L_{0}\left(t, x, v, a\right) - 2 \, s \frac{\partial^{3}}{\partial x(\partial a)^{2}}L_{0}\left(t, x, v, a\right) + 6 \, a \frac{\partial^{3}}{\partial t\partial x\partial v}L_{1}\left(t, x, v, a\right) + 4 \, s \frac{\partial^{3}}{\partial t\partial x\partial a}L_{1}\left(t, x, v, a\right) + 3 \, a \frac{\partial^{2}}{(\partial x)^{2}}L_{1}\left(t, x, v, a\right) + 4 \, s \frac{\partial^{2}}{\partial x\partial v}L_{1}\left(t, x, v, a\right) - 2 \, \frac{\partial^{3}}{\partial t\partial x\partial a}L_{0}\left(t, x, v, a\right) + \frac{\partial^{2}}{\partial x\partial v}L_{0}\left(t, x, v, a\right) + 3 \, \frac{\partial^{3}}{(\partial t)^{2}\partial x}L_{1}\left(t, x, v, a\right)\right)} v - 2 \, a \frac{\partial^{3}}{\partial t\partial v\partial a}L_{0}\left(t, x, v, a\right) - 2 \, s \frac{\partial^{3}}{\partial t(\partial a)^{2}}L_{0}\left(t, x, v, a\right) - a \frac{\partial^{2}}{\partial x\partial a}L_{0}\left(t, x, v, a\right) + a \frac{\partial^{2}}{(\partial v)^{2}}L_{0}\left(t, x, v, a\right) + 3 \, a \frac{\partial^{3}}{(\partial t)^{2}\partial v}L_{1}\left(t, x, v, a\right) + 2 \, s \frac{\partial^{3}}{(\partial t)^{2}\partial a}L_{1}\left(t, x, v, a\right) + 3 \, a \frac{\partial^{2}}{\partial t\partial x}L_{1}\left(t, x, v, a\right) + 4 \, s \frac{\partial^{2}}{\partial t\partial v}L_{1}\left(t, x, v, a\right) - \frac{\partial^{3}}{(\partial t)^{2}\partial a}L_{0}\left(t, x, v, a\right) + \frac{\partial^{2}}{\partial t\partial v}L_{0}\left(t, x, v, a\right) - \frac{\partial}{\partial x}L_{0}\left(t, x, v, a\right) + \frac{\partial^{3}}{(\partial t)^{3}}L_{1}\left(t, x, v, a\right)}{v \frac{\partial^{2}}{\partial x\partial a}L_{1}\left(t, x, v, a\right) + a \frac{\partial^{2}}{\partial v\partial a}L_{1}\left(t, x, v, a\right) - \frac{\partial^{2}}{(\partial a)^{2}}L_{0}\left(t, x, v, a\right) + \frac{\partial^{2}}{\partial t\partial a}L_{1}\left(t, x, v, a\right) + 2 \, \frac{\partial}{\partial v}L_{1}\left(t, x, v, a\right)}$

Second Equation of Lagrange

(Dt(L)-(N(L)-(p*a+q*s+r*f)-(v*N(p)+a*N(q)+s*N(r)))).display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & v^{4} \frac{\partial^4\,L}{\partial x ^ 3\partial s} + {\left(3 \, a \frac{\partial^4\,L}{\partial x ^ 2\partial v\partial s} + 3 \, s \frac{\partial^4\,L}{\partial x ^ 2\partial a\partial s} + 3 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x ^ 2\partial s ^ 2} + 3 \, \frac{\partial^3\,L}{\partial x\partial s ^ 2} \frac{\partial\,f}{\partial x} + \frac{\partial^2\,L}{\partial s ^ 2} \frac{\partial^2\,f}{\partial x ^ 2} + 3 \, \frac{\partial^4\,L}{\partial t\partial x ^ 2\partial s} - \frac{\partial^3\,L}{\partial x ^ 2\partial a}\right)} v^{3} + {\left(3 \, a^{2} \frac{\partial^4\,L}{\partial x\partial v ^ 2\partial s} + 6 \, a s \frac{\partial^4\,L}{\partial x\partial v\partial a\partial s} + 6 \, a f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x\partial v\partial s ^ 2} + 3 \, s^{2} \frac{\partial^4\,L}{\partial x\partial a ^ 2\partial s} + 6 \, s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial x\partial a\partial s ^ 2} + 3 \, f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial x\partial s ^ 3} + 3 \, a \frac{\partial^3\,L}{\partial v\partial s ^ 2} \frac{\partial\,f}{\partial x} + 3 \, s \frac{\partial^3\,L}{\partial a\partial s ^ 2} \frac{\partial\,f}{\partial x} + 3 \, f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial s ^ 3} \frac{\partial\,f}{\partial x} + 6 \, a \frac{\partial^4\,L}{\partial t\partial x\partial v\partial s} + 6 \, s \frac{\partial^4\,L}{\partial t\partial x\partial a\partial s} + 6 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial x\partial s ^ 2} + 3 \, a \frac{\partial^3\,L}{\partial x ^ 2\partial s} - 2 \, a \frac{\partial^3\,L}{\partial x\partial v\partial a} + 3 \, s \frac{\partial^3\,L}{\partial x\partial v\partial s} - 2 \, s \frac{\partial^3\,L}{\partial x\partial a ^ 2} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial x\partial a\partial s} + 3 \, {\left(a \frac{\partial\,f}{\partial v} + s \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial x\partial s ^ 2} + {\left(2 \, a \frac{\partial^2\,f}{\partial x\partial v} + 2 \, s \frac{\partial^2\,f}{\partial x\partial a} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^2\,f}{\partial x\partial s} + \frac{\partial\,f}{\partial x} \frac{\partial\,f}{\partial s} + 2 \, \frac{\partial^2\,f}{\partial t\partial x}\right)} \frac{\partial^2\,L}{\partial s ^ 2} + 3 \, \frac{\partial^3\,L}{\partial t\partial s ^ 2} \frac{\partial\,f}{\partial x} + 3 \, \frac{\partial^4\,L}{\partial t ^ 2\partial x\partial s} - 2 \, \frac{\partial^3\,L}{\partial t\partial x\partial a} + \frac{\partial^2\,L}{\partial x\partial v}\right)} v^{2} + {\left(a^{3} \frac{\partial^4\,L}{\partial v ^ 3\partial s} + 3 \, a^{2} s \frac{\partial^4\,L}{\partial v ^ 2\partial a\partial s} + 3 \, a^{2} f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial v ^ 2\partial s ^ 2} + 3 \, a s^{2} \frac{\partial^4\,L}{\partial v\partial a ^ 2\partial s} + 6 \, a s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial v\partial a\partial s ^ 2} + 3 \, a f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial v\partial s ^ 3} + s^{3} \frac{\partial^4\,L}{\partial a ^ 3\partial s} + 3 \, s^{2} f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial a ^ 2\partial s ^ 2} + 3 \, s f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial a\partial s ^ 3} + f\left(t, x, v, a, s\right)^{3} \frac{\partial^4\,L}{\partial s ^ 4} + 3 \, a^{2} \frac{\partial^4\,L}{\partial t\partial v ^ 2\partial s} + 6 \, a s \frac{\partial^4\,L}{\partial t\partial v\partial a\partial s} + 6 \, a f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial v\partial s ^ 2} + 3 \, s^{2} \frac{\partial^4\,L}{\partial t\partial a ^ 2\partial s} + 6 \, s f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t\partial a\partial s ^ 2} + 3 \, f\left(t, x, v, a, s\right)^{2} \frac{\partial^4\,L}{\partial t\partial s ^ 3} + 3 \, a^{2} \frac{\partial^3\,L}{\partial x\partial v\partial s} + 3 \, a s \frac{\partial^3\,L}{\partial x\partial a\partial s} + 3 \, a f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial x\partial s ^ 2} - a^{2} \frac{\partial^3\,L}{\partial v ^ 2\partial a} + 3 \, a s \frac{\partial^3\,L}{\partial v ^ 2\partial s} - 2 \, a s \frac{\partial^3\,L}{\partial v\partial a ^ 2} - s^{2} \frac{\partial^3\,L}{\partial a ^ 3} + s f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial a ^ 2\partial s} + 3 \, a \frac{\partial^4\,L}{\partial t ^ 2\partial v\partial s} + 3 \, s \frac{\partial^4\,L}{\partial t ^ 2\partial a\partial s} + 3 \, f\left(t, x, v, a, s\right) \frac{\partial^4\,L}{\partial t ^ 2\partial s ^ 2} + 3 \, a \frac{\partial^3\,L}{\partial t\partial x\partial s} - 2 \, a \frac{\partial^3\,L}{\partial t\partial v\partial a} + 3 \, s \frac{\partial^3\,L}{\partial t\partial v\partial s} - 2 \, s \frac{\partial^3\,L}{\partial t\partial a ^ 2} + f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial t\partial a\partial s} + 3 \, {\left(a \frac{\partial\,f}{\partial v} + s \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial t\partial s ^ 2} - a \frac{\partial^2\,L}{\partial x\partial a} + s \frac{\partial^2\,L}{\partial x\partial s} + a \frac{\partial^2\,L}{\partial v ^ 2} + {\left(3 \, s^{2} + a f\left(t, x, v, a, s\right)\right)} \frac{\partial^3\,L}{\partial v\partial a\partial s} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^2\,L}{\partial v\partial s} + 3 \, {\left(a^{2} \frac{\partial\,f}{\partial v} + a s \frac{\partial\,f}{\partial a} + a f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + s f\left(t, x, v, a, s\right) + a \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial v\partial s ^ 2} - f\left(t, x, v, a, s\right) \frac{\partial^2\,L}{\partial a ^ 2} + {\left(3 \, a s \frac{\partial\,f}{\partial v} + 3 \, s^{2} \frac{\partial\,f}{\partial a} + 3 \, s f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + 2 \, f\left(t, x, v, a, s\right)^{2} + 3 \, s \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial a\partial s ^ 2} + {\left(a^{2} \frac{\partial^2\,f}{\partial v ^ 2} + 2 \, a s \frac{\partial^2\,f}{\partial v\partial a} + 2 \, a f\left(t, x, v, a, s\right) \frac{\partial^2\,f}{\partial v\partial s} + s^{2} \frac{\partial^2\,f}{\partial a ^ 2} + 2 \, s f\left(t, x, v, a, s\right) \frac{\partial^2\,f}{\partial a\partial s} + s \frac{\partial\,f}{\partial a} \frac{\partial\,f}{\partial s} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s}^{2} + f\left(t, x, v, a, s\right)^{2} \frac{\partial^2\,f}{\partial s ^ 2} + 2 \, a \frac{\partial^2\,f}{\partial t\partial v} + 2 \, s \frac{\partial^2\,f}{\partial t\partial a} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^2\,f}{\partial t\partial s} + a \frac{\partial\,f}{\partial x} + {\left(a \frac{\partial\,f}{\partial s} + s\right)} \frac{\partial\,f}{\partial v} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial a} + \frac{\partial\,f}{\partial t} \frac{\partial\,f}{\partial s} + \frac{\partial^2\,f}{\partial t ^ 2}\right)} \frac{\partial^2\,L}{\partial s ^ 2} + 3 \, {\left(a f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial v} + s f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right)^{2} \frac{\partial\,f}{\partial s} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial t}\right)} \frac{\partial^3\,L}{\partial s ^ 3} + \frac{\partial^4\,L}{\partial t ^ 3\partial s} - \frac{\partial^3\,L}{\partial t ^ 2\partial a} + \frac{\partial^2\,L}{\partial t\partial v} - \frac{\partial\,L}{\partial x}\right)} v \end{array}$

Multiplying by $v$

v*(N(p)-Dx(L)) == (Dt(L)-(N(L)-(p*a+q*s+r*f)-(v*N(p)+a*N(q)+s*N(r))))

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

Checking the calculations from the paper

L = M.scalar_field(function('L')(*list(coord)))p = M.scalar_field(function('p')(*list(coord)))q = M.scalar_field(function('q')(*list(coord)))r = M.scalar_field(function('r')(*list(coord)))t=M.scalar_field(t)x=M.scalar_field(x)v=M.scalar_field(v)a=M.scalar_field(a)s=M.scalar_field(s)

Action differential Form

alpha = L*dt + p*(dx-v*dt) + q*(dv-a*dt) + r*(da-s*dt)alpha.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\left( -v p\left(t, x, v, a, s\right) - a q\left(t, x, v, a, s\right) - s r\left(t, x, v, a, s\right) + L\left(t, x, v, a, s\right) \right) \mathrm{d} t + p\left(t, x, v, a, s\right) \mathrm{d} x + q\left(t, x, v, a, s\right) \mathrm{d} v + r\left(t, x, v, a, s\right) \mathrm{d} a$

alpha == L*dt+p*dx+q*dv+r*da-(p*v+q*a+r*s)*dt

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

d(alpha).display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\left( v \frac{\partial\,p}{\partial x} + a \frac{\partial\,q}{\partial x} + s \frac{\partial\,r}{\partial x} - \frac{\partial\,L}{\partial x} + \frac{\partial\,p}{\partial t} \right) \mathrm{d} t\wedge \mathrm{d} x + \left( v \frac{\partial\,p}{\partial v} + a \frac{\partial\,q}{\partial v} + s \frac{\partial\,r}{\partial v} + p\left(t, x, v, a, s\right) - \frac{\partial\,L}{\partial v} + \frac{\partial\,q}{\partial t} \right) \mathrm{d} t\wedge \mathrm{d} v + \left( v \frac{\partial\,p}{\partial a} + a \frac{\partial\,q}{\partial a} + s \frac{\partial\,r}{\partial a} + q\left(t, x, v, a, s\right) - \frac{\partial\,L}{\partial a} + \frac{\partial\,r}{\partial t} \right) \mathrm{d} t\wedge \mathrm{d} a + \left( v \frac{\partial\,p}{\partial s} + a \frac{\partial\,q}{\partial s} + s \frac{\partial\,r}{\partial s} + r\left(t, x, v, a, s\right) - \frac{\partial\,L}{\partial s} \right) \mathrm{d} t\wedge \mathrm{d} s + \left( -\frac{\partial\,p}{\partial v} + \frac{\partial\,q}{\partial x} \right) \mathrm{d} x\wedge \mathrm{d} v + \left( -\frac{\partial\,p}{\partial a} + \frac{\partial\,r}{\partial x} \right) \mathrm{d} x\wedge \mathrm{d} a -\frac{\partial\,p}{\partial s} \mathrm{d} x\wedge \mathrm{d} s + \left( -\frac{\partial\,q}{\partial a} + \frac{\partial\,r}{\partial v} \right) \mathrm{d} v\wedge \mathrm{d} a -\frac{\partial\,q}{\partial s} \mathrm{d} v\wedge \mathrm{d} s -\frac{\partial\,r}{\partial s} \mathrm{d} a\wedge \mathrm{d} s$

d(alpha) == d(L).wedge(dt) + d(p).wedge(dx) + d(q).wedge(dv) + d(r).wedge(da) - d(p*v + q*a + r*s).wedge(dt)

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

ev(alpha)(N)==L

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

Omega = -(p*dx+q*dv+r*da).wedge(d(t)); Omega.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}p\left(t, x, v, a, s\right) \mathrm{d} t\wedge \mathrm{d} x + q\left(t, x, v, a, s\right) \mathrm{d} t\wedge \mathrm{d} v + r\left(t, x, v, a, s\right) \mathrm{d} t\wedge \mathrm{d} a$

alpha == L*dt + ev(N)(Omega)

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

Equation of Motion ($E=0$)

E = ev(N)(d(alpha))E.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\left( -v^{2} \frac{\partial\,p}{\partial x} - a^{2} \frac{\partial\,q}{\partial v} - a s \frac{\partial\,q}{\partial a} - a f\left(t, x, v, a, s\right) \frac{\partial\,q}{\partial s} - a s \frac{\partial\,r}{\partial v} - s^{2} \frac{\partial\,r}{\partial a} - s f\left(t, x, v, a, s\right) \frac{\partial\,r}{\partial s} - {\left(a \frac{\partial\,p}{\partial v} + s \frac{\partial\,p}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,p}{\partial s} + a \frac{\partial\,q}{\partial x} + s \frac{\partial\,r}{\partial x} - \frac{\partial\,L}{\partial x} + \frac{\partial\,p}{\partial t}\right)} v - a p\left(t, x, v, a, s\right) - s q\left(t, x, v, a, s\right) - f\left(t, x, v, a, s\right) r\left(t, x, v, a, s\right) + a \frac{\partial\,L}{\partial v} + s \frac{\partial\,L}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,L}{\partial s} - a \frac{\partial\,q}{\partial t} - s \frac{\partial\,r}{\partial t} \right) \mathrm{d} t + \left( v \frac{\partial\,p}{\partial x} + a \frac{\partial\,p}{\partial v} + s \frac{\partial\,p}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,p}{\partial s} - \frac{\partial\,L}{\partial x} + \frac{\partial\,p}{\partial t} \right) \mathrm{d} x + \left( v \frac{\partial\,q}{\partial x} + a \frac{\partial\,q}{\partial v} + s \frac{\partial\,q}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,q}{\partial s} + p\left(t, x, v, a, s\right) - \frac{\partial\,L}{\partial v} + \frac{\partial\,q}{\partial t} \right) \mathrm{d} v + \left( v \frac{\partial\,r}{\partial x} + a \frac{\partial\,r}{\partial v} + s \frac{\partial\,r}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,r}{\partial s} + q\left(t, x, v, a, s\right) - \frac{\partial\,L}{\partial a} + \frac{\partial\,r}{\partial t} \right) \mathrm{d} a + \left( r\left(t, x, v, a, s\right) - \frac{\partial\,L}{\partial s} \right) \mathrm{d} s$

Rewriting it in various ways.

E == N(L)*dt - N(t) * d(L) + N(p)*dx - N(x)*d(p) + N(q)*dv - N(v)*d(q) + N(r)*da - N(a)*d(r) - N(p*v+q*a+r*s)*dt + N(t)*d(p*v+q*a+r*s)

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

E == N(L)*dt - d(L) + N(p)*dx - v*d(p) + N(q)*dv - a*d(q) + N(r)*da - s*d(r) - N(p*v+q*a+r*s)*dt + d(p*v+q*a+r*s)

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

E == N(L-p*v-q*a-r*s)*dt - d(L - p*v - q*a- r*s) - v*d(p) - a*d(q) - s*d(r) + N(p)*dx + N(q)*dv + N(r)*da

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

d(p*v) == v*d(p) + p*d(v)

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

d(q*a) == a*d(q) + q*d(a)

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

d(r*s) == s*d(r) + r*d(s)

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

E == N(L - p*v - q*a - r*s)*dt - d(L) + p*dv + q*da + r*ds + N(p)*dx + N(q)*dv + N(r)*da

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

N(p*v) == p*N(v) + v*N(p)

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

E == N(L)*dt - (p*a+q*s+r*f)*dt -(v*N(p) + a*N(q) + s*N(r) )*dt - d(L) + p*dv + q*da + r*ds + N(p)*dx + N(q)*dv + N(r)*da

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

E == ( N(L) - (p*a + q*s + r*f) - (v*N(p) + a*N(q) + s*N(r)))*dt + N(p)*dx + (N(q)+p)*dv + (N(r)+q)*da + r*ds - d(L)

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

d(L) == Dt(L)*dt + Dx(L)*dx + Dv(L)*dv + Da(L)*da + Ds(L)*ds

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

r=Ds(L); r.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & \frac{\partial\,L}{\partial s} \end{array}$

q=Da(L)-N(r); q.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & -v \frac{\partial^2\,L}{\partial x\partial s} - a \frac{\partial^2\,L}{\partial v\partial s} - s \frac{\partial^2\,L}{\partial a\partial s} - f\left(t, x, v, a, s\right) \frac{\partial^2\,L}{\partial s ^ 2} - \frac{\partial^2\,L}{\partial t\partial s} + \frac{\partial\,L}{\partial a} \end{array}$

p=Dv(L)-N(q); p.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & v^{2} \frac{\partial^3\,L}{\partial x ^ 2\partial s} + a^{2} \frac{\partial^3\,L}{\partial v ^ 2\partial s} + 2 \, a s \frac{\partial^3\,L}{\partial v\partial a\partial s} + 2 \, a f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial v\partial s ^ 2} + s^{2} \frac{\partial^3\,L}{\partial a ^ 2\partial s} + 2 \, s f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial a\partial s ^ 2} + f\left(t, x, v, a, s\right)^{2} \frac{\partial^3\,L}{\partial s ^ 3} + {\left(2 \, a \frac{\partial^3\,L}{\partial x\partial v\partial s} + 2 \, s \frac{\partial^3\,L}{\partial x\partial a\partial s} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial x\partial s ^ 2} + \frac{\partial^2\,L}{\partial s ^ 2} \frac{\partial\,f}{\partial x} + 2 \, \frac{\partial^3\,L}{\partial t\partial x\partial s} - \frac{\partial^2\,L}{\partial x\partial a}\right)} v + 2 \, a \frac{\partial^3\,L}{\partial t\partial v\partial s} + 2 \, s \frac{\partial^3\,L}{\partial t\partial a\partial s} + 2 \, f\left(t, x, v, a, s\right) \frac{\partial^3\,L}{\partial t\partial s ^ 2} + a \frac{\partial^2\,L}{\partial x\partial s} - a \frac{\partial^2\,L}{\partial v\partial a} + s \frac{\partial^2\,L}{\partial v\partial s} - s \frac{\partial^2\,L}{\partial a ^ 2} + {\left(a \frac{\partial\,f}{\partial v} + s \frac{\partial\,f}{\partial a} + f\left(t, x, v, a, s\right) \frac{\partial\,f}{\partial s} + \frac{\partial\,f}{\partial t}\right)} \frac{\partial^2\,L}{\partial s ^ 2} + \frac{\partial^3\,L}{\partial t ^ 2\partial s} - \frac{\partial^2\,L}{\partial t\partial a} + \frac{\partial\,L}{\partial v} \end{array}$

For example: The Schiff and Poirier Lagrangian

hbar = var('hbar',latex_name='\hbar')m = var('m')V = M.scalar_field(function('V')(var('x')))Lp = 1/2*m*v^2  - V - hbar^2/4/m*(s/v^3-5/2*a^2/v^4); Lp.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & \frac{4 \, m^{2} v^{6} - 8 \, m v^{4} V\left(x\right) + 5 \, a^{2} {\hbar}^{2} - 2 \, {\hbar}^{2} s v}{8 \, m v^{4}} \end{array}$

Lp1 = Ds(Lp); Lp1.display()

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & -\frac{{\hbar}^{2}}{4 \, m v^{3}} \end{array}$

Lp0 = Lp - s * Lp1; Lp0.display()

Lp == Lp0 + s*Lp1

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

eq1p = eq1.expr().substitute_function(L.expr().operator(),Lp.expr().function(*list(coord))); eq1p

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, a {\left(\frac{6 \, {\left(5 \, m^{2} v^{4} - 4 \, m v^{2} V\left(x\right)\right)}}{m v^{4}} - \frac{4 \, {\left(12 \, m^{2} v^{5} - 16 \, m v^{3} V\left(x\right) - {\hbar}^{2} s\right)}}{m v^{5}} + \frac{5 \, {\left(4 \, m^{2} v^{6} - 8 \, m v^{4} V\left(x\right) + 5 \, a^{2} {\hbar}^{2} - 2 \, {\hbar}^{2} s v\right)}}{m v^{6}}\right)} - \frac{10 \, a^{3} {\hbar}^{2}}{m v^{6}} + \frac{a {\hbar}^{2} s}{m v^{5}} + \frac{{\hbar}^{2} f\left(t, x, v, a, s\right)}{4 \, m v^{4}} + \frac{\partial}{\partial x}V\left(x\right)$

solve(eq1p,f.expr())

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[f\left(t, x, v, a, s\right) = -\frac{2 \, {\left(2 \, a m^{2} v^{6} + 2 \, m v^{6} \frac{\partial}{\partial x}V\left(x\right) + 5 \, a^{3} {\hbar}^{2} - 4 \, a {\hbar}^{2} s v\right)}}{{\hbar}^{2} v^{2}}\right]$