Contact
CoCalc Logo Icon
StoreFeaturesDocsShareSupport News Sign UpSign In
| Download

Vanilla version

Views: 126
Kernel: SageMath (stable)

Higher-Order Equation of Motion

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

Related worksheet: https://cloud.sagemath.com/projects/b04b5777-e269-4c8f-a4b8-b21dbe1c93c6/files/Two%20Equations%20of%20Lagrange.sagews

Calculations are done using SageManifolds.

%display typeset from sage.manifolds.utilities import exterior_derivative as d def 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()
MR(t,x,v,a,s)L(t,x,v,a,s)\renewcommand{\Bold}[1]{\mathbf{#1}}\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

ff 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()
MR(t,x,v,a,s)f(t,x,v,a,s)\renewcommand{\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) \end{array}
N = Dt + v*Dx + a*Dv + s*Da + f*Ds; N.display()
t+vx+av+sa+f(t,x,v,a,s)s\renewcommand{\Bold}[1]{\mathbf{#1}}\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()
MR(t,x,v,a,s)Ls\renewcommand{\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()
MR(t,x,v,a,s)v2Lxsa2Lvss2Lasf(t,x,v,a,s)2Ls22Lts+La\renewcommand{\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()
MR(t,x,v,a,s)v23Lx2s+a23Lv2s+2as3Lvas+2af(t,x,v,a,s)3Lvs2+s23La2s+2sf(t,x,v,a,s)3Las2+f(t,x,v,a,s)23Ls3+(2a3Lxvs+2s3Lxas+2f(t,x,v,a,s)3Lxs2+2Ls2fx+23Ltxs2Lxa)v+2a3Ltvs+2s3Ltas+2f(t,x,v,a,s)3Lts2+a2Lxsa2Lva+s2Lvss2La2+(afv+sfa+f(t,x,v,a,s)fs+ft)2Ls2+3Lt2s2Lta+Lv\renewcommand{\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}
eq1 = (N(p)-Dx(L)); eq1.display()
MR(t,x,v,a,s)v34Lx3s+a34Lv3s+3a2s4Lv2as+3a2f(t,x,v,a,s)4Lv2s2+3as24Lva2s+6asf(t,x,v,a,s)4Lvas2+3af(t,x,v,a,s)24Lvs3+s34La3s+3s2f(t,x,v,a,s)4La2s2+3sf(t,x,v,a,s)24Las3+f(t,x,v,a,s)34Ls4+(3a4Lx2vs+3s4Lx2as+3f(t,x,v,a,s)4Lx2s2+33Lxs2fx+2Ls22fx2+34Ltx2s3Lx2a)v2+3a24Ltv2s+6as4Ltvas+6af(t,x,v,a,s)4Ltvs2+3s24Lta2s+6sf(t,x,v,a,s)4Ltas2+3f(t,x,v,a,s)24Lts3+3a23Lxvs+3as3Lxas+3af(t,x,v,a,s)3Lxs2a23Lv2a+3as3Lv2s2as3Lva2s23La3+sf(t,x,v,a,s)3La2s+(3a24Lxv2s+6as4Lxvas+6af(t,x,v,a,s)4Lxvs2+3s24Lxa2s+6sf(t,x,v,a,s)4Lxas2+3f(t,x,v,a,s)24Lxs3+3a3Lvs2fx+3s3Las2fx+3f(t,x,v,a,s)3Ls3fx+6a4Ltxvs+6s4Ltxas+6f(t,x,v,a,s)4Ltxs2+3a3Lx2s2a3Lxva+3s3Lxvs2s3Lxa2+f(t,x,v,a,s)3Lxas+3(afv+sfa+f(t,x,v,a,s)fs+ft)3Lxs2+(2a2fxv+2s2fxa+2f(t,x,v,a,s)2fxs+fxfs+22ftx)2Ls2+33Lts2fx+34Lt2xs23Ltxa+2Lxv)v+3a4Lt2vs+3s4Lt2as+3f(t,x,v,a,s)4Lt2s2+3a3Ltxs2a3Ltva+3s3Ltvs2s3Lta2+f(t,x,v,a,s)3Ltas+3(afv+sfa+f(t,x,v,a,s)fs+ft)3Lts2a2Lxa+s2Lxs+a2Lv2+(3s2+af(t,x,v,a,s))3Lvas+2f(t,x,v,a,s)2Lvs+3(a2fv+asfa+af(t,x,v,a,s)fs+sf(t,x,v,a,s)+aft)3Lvs2f(t,x,v,a,s)2La2+(3asfv+3s2fa+3sf(t,x,v,a,s)fs+2f(t,x,v,a,s)2+3sft)3Las2+(a22fv2+2as2fva+2af(t,x,v,a,s)2fvs+s22fa2+2sf(t,x,v,a,s)2fas+sfafs+f(t,x,v,a,s)fs2+f(t,x,v,a,s)22fs2+2a2ftv+2s2fta+2f(t,x,v,a,s)2fts+afx+(afs+s)fv+f(t,x,v,a,s)fa+ftfs+2ft2)2Ls2+3(af(t,x,v,a,s)fv+sf(t,x,v,a,s)fa+f(t,x,v,a,s)2fs+f(t,x,v,a,s)ft)3Ls3+4Lt3s3Lt2a+2LtvLx\renewcommand{\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} + \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())
116\renewcommand{\Bold}[1]{\mathbf{#1}}116

We want to solve this equation (eq1 = 0) for ff in terms of LL.

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
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}

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

N(N(f)).display()
MR(t,x,v,a,s)v22fx2+a22fv2+2as2fva+2af(t,x,v,a,s)2fvs+s22fa2+2sf(t,x,v,a,s)2fas+sfafs+f(t,x,v,a,s)fs2+f(t,x,v,a,s)22fs2+(2a2fxv+2s2fxa+2f(t,x,v,a,s)2fxs+fxfs+22ftx)v+2a2ftv+2s2fta+2f(t,x,v,a,s)2fts+afx+(afs+s)fv+f(t,x,v,a,s)fa+ftfs+2ft2\renewcommand{\Bold}[1]{\mathbf{#1}}\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$^\textit{nd}orderin order in farisefromtheLiebnizruleappliedtothedifferentialoperator arise from the Liebniz rule applied to the differential operator N^2$ on the term Ds fDs\ f in N(Ds(L))N(Ds(L)). For example

t2 = eq1.expr().coefficient(diff(f.expr(),t,t))*diff(f.expr(),t,t); M.scalar_field(t2).display()
MR(t,x,v,a,s)2Ls22ft2\renewcommand{\Bold}[1]{\mathbf{#1}}\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()
MR(t,x,v,a,s)v22Ls22fx2+(2a2fxv+2s2fxa+2f(t,x,v,a,s)2fxs+fxfs+22ftx)v2Ls2+(a22fv2+2as2fva+2af(t,x,v,a,s)2fvs+s22fa2+2sf(t,x,v,a,s)2fas+sfafs+f(t,x,v,a,s)fs2+f(t,x,v,a,s)22fs2+2a2ftv+2s2fta+2f(t,x,v,a,s)2fts+afx+(afs+s)fv+f(t,x,v,a,s)fa+ftfs+2ft2)2Ls2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, v, a, s\right) & \longmapsto & v^{2} \frac{\partial^2\,L}{\partial s ^ 2} \frac{\partial^2\,f}{\partial x ^ 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 \frac{\partial^2\,L}{\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} \end{array}

After removing the term c2c2 above, the remaining terms are at most first-order in ff.

eq1a = eq1-c2; eq1a.display()
MR(t,x,v,a,s)v34Lx3s+a34Lv3s+3a2s4Lv2as+3a2f(t,x,v,a,s)4Lv2s2+3as24Lva2s+6asf(t,x,v,a,s)4Lvas2+3af(t,x,v,a,s)24Lvs3+s34La3s+3s2f(t,x,v,a,s)4La2s2+3sf(t,x,v,a,s)24Las3+f(t,x,v,a,s)34Ls4+(3a4Lx2vs+3s4Lx2as+3f(t,x,v,a,s)4Lx2s2+33Lxs2fx+34Ltx2s3Lx2a)v2+3a24Ltv2s+6as4Ltvas+6af(t,x,v,a,s)4Ltvs2+3s24Lta2s+6sf(t,x,v,a,s)4Ltas2+3f(t,x,v,a,s)24Lts3+3a23Lxvs+3as3Lxas+3af(t,x,v,a,s)3Lxs2a23Lv2a+3as3Lv2s2as3Lva2s23La3+sf(t,x,v,a,s)3La2s+(3a24Lxv2s+6as4Lxvas+6af(t,x,v,a,s)4Lxvs2+3s24Lxa2s+6sf(t,x,v,a,s)4Lxas2+3f(t,x,v,a,s)24Lxs3+3a3Lvs2fx+3s3Las2fx+3f(t,x,v,a,s)3Ls3fx+6a4Ltxvs+6s4Ltxas+6f(t,x,v,a,s)4Ltxs2+3a3Lx2s2a3Lxva+3s3Lxvs2s3Lxa2+f(t,x,v,a,s)3Lxas+3(afv+sfa+f(t,x,v,a,s)fs+ft)3Lxs2+33Lts2fx+34Lt2xs23Ltxa+2Lxv)v+3a4Lt2vs+3s4Lt2as+3f(t,x,v,a,s)4Lt2s2+3a3Ltxs2a3Ltva+3s3Ltvs2s3Lta2+f(t,x,v,a,s)3Ltas+3(afv+sfa+f(t,x,v,a,s)fs+ft)3Lts2a2Lxa+s2Lxs+a2Lv2+(3s2+af(t,x,v,a,s))3Lvas+2f(t,x,v,a,s)2Lvs+3(a2fv+asfa+af(t,x,v,a,s)fs+sf(t,x,v,a,s)+aft)3Lvs2f(t,x,v,a,s)2La2+(3asfv+3s2fa+3sf(t,x,v,a,s)fs+2f(t,x,v,a,s)2+3sft)3Las2+3(af(t,x,v,a,s)fv+sf(t,x,v,a,s)fa+f(t,x,v,a,s)2fs+f(t,x,v,a,s)ft)3Ls3+4Lt3s3Lt2a+2LtvLx\renewcommand{\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 NN applied to the term Ds fDs\ f in N(Da(L))N(Da(L)).

N(f).display()
MR(t,x,v,a,s)vfx+afv+sfa+f(t,x,v,a,s)fs+ft\renewcommand{\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()
MR(t,x,v,a,s)3(v3Lxs2+a3Lvs2+s3Las2+f(t,x,v,a,s)3Ls3+3Lts2)ft\renewcommand{\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()
MR(t,x,v,a,s)3(v23Lxs2+(a3Lvs2+s3Las2+f(t,x,v,a,s)3Ls3+3Lts2)v)fx\renewcommand{\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()
MR(t,x,v,a,s)3(av3Lxs2+a23Lvs2+as3Las2+af(t,x,v,a,s)3Ls3+a3Lts2)fv\renewcommand{\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()
MR(t,x,v,a,s)3(sv3Lxs2+as3Lvs2+s23Las2+sf(t,x,v,a,s)3Ls3+s3Lts2)fa\renewcommand{\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()
MR(t,x,v,a,s)3(vf(t,x,v,a,s)3Lxs2+af(t,x,v,a,s)3Lvs2+sf(t,x,v,a,s)3Las2+f(t,x,v,a,s)23Ls3+f(t,x,v,a,s)3Lts2)fs\renewcommand{\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 NN to these coefficients.

c1 = 3*N(Ds(Ds(L)))*N(f); c1.display()
MR(t,x,v,a,s)3v23Lxs2fx+3(a3Lvs2fx+s3Las2fx+f(t,x,v,a,s)3Ls3fx+(afv+sfa+f(t,x,v,a,s)fs+ft)3Lxs2+3Lts2fx)v+3(afv+sfa+f(t,x,v,a,s)fs+ft)3Lts2+3(a2fv+asfa+af(t,x,v,a,s)fs+aft)3Lvs2+3(asfv+s2fa+sf(t,x,v,a,s)fs+sft)3Las2+3(af(t,x,v,a,s)fv+sf(t,x,v,a,s)fa+f(t,x,v,a,s)2fs+f(t,x,v,a,s)ft)3Ls3\renewcommand{\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)
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}

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

eq1b = eq1a-c1; eq1b.display()
MR(t,x,v,a,s)v34Lx3s+a34Lv3s+3a2s4Lv2as+3a2f(t,x,v,a,s)4Lv2s2+3as24Lva2s+6asf(t,x,v,a,s)4Lvas2+3af(t,x,v,a,s)24Lvs3+s34La3s+3s2f(t,x,v,a,s)4La2s2+3sf(t,x,v,a,s)24Las3+f(t,x,v,a,s)34Ls4+(3a4Lx2vs+3s4Lx2as+3f(t,x,v,a,s)4Lx2s2+34Ltx2s3Lx2a)v2+3a24Ltv2s+6as4Ltvas+6af(t,x,v,a,s)4Ltvs2+3s24Lta2s+6sf(t,x,v,a,s)4Ltas2+3f(t,x,v,a,s)24Lts3+3a23Lxvs+3as3Lxas+3af(t,x,v,a,s)3Lxs2a23Lv2a+3as3Lv2s2as3Lva2+3sf(t,x,v,a,s)3Lvs2s23La3+sf(t,x,v,a,s)3La2s+2f(t,x,v,a,s)23Las2+(3a24Lxv2s+6as4Lxvas+6af(t,x,v,a,s)4Lxvs2+3s24Lxa2s+6sf(t,x,v,a,s)4Lxas2+3f(t,x,v,a,s)24Lxs3+6a4Ltxvs+6s4Ltxas+6f(t,x,v,a,s)4Ltxs2+3a3Lx2s2a3Lxva+3s3Lxvs2s3Lxa2+f(t,x,v,a,s)3Lxas+34Lt2xs23Ltxa+2Lxv)v+3a4Lt2vs+3s4Lt2as+3f(t,x,v,a,s)4Lt2s2+3a3Ltxs2a3Ltva+3s3Ltvs2s3Lta2+f(t,x,v,a,s)3Ltasa2Lxa+s2Lxs+a2Lv2+(3s2+af(t,x,v,a,s))3Lvas+2f(t,x,v,a,s)2Lvsf(t,x,v,a,s)2La2+4Lt3s3Lt2a+2LtvLx\renewcommand{\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 ff remain. The number of terms is:

len(eq1b.expr().expand())
68\renewcommand{\Bold}[1]{\mathbf{#1}}68

The term of highest degree comes from N3N^3 applied to Ds(L)Ds(L):

t0 = eq1b.expr().coefficient(f.expr()^3)*f.expr()^3; M.scalar_field(t0.expand()).display()
MR(t,x,v,a,s)f(t,x,v,a,s)34Ls4\renewcommand{\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()
MR(t,x,v,a,s)f(t,x,v,a,s)34Ls4\renewcommand{\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}

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

eq1c = eq1b-cf3; eq1c.display()
MR(t,x,v,a,s)v34Lx3s+a34Lv3s+3a2s4Lv2as+3a2f(t,x,v,a,s)4Lv2s2+3as24Lva2s+6asf(t,x,v,a,s)4Lvas2+3af(t,x,v,a,s)24Lvs3+s34La3s+3s2f(t,x,v,a,s)4La2s2+3sf(t,x,v,a,s)24Las3+(3a4Lx2vs+3s4Lx2as+3f(t,x,v,a,s)4Lx2s2+34Ltx2s3Lx2a)v2+3a24Ltv2s+6as4Ltvas+6af(t,x,v,a,s)4Ltvs2+3s24Lta2s+6sf(t,x,v,a,s)4Ltas2+3f(t,x,v,a,s)24Lts3+3a23Lxvs+3as3Lxas+3af(t,x,v,a,s)3Lxs2a23Lv2a+3as3Lv2s2as3Lva2+3sf(t,x,v,a,s)3Lvs2s23La3+sf(t,x,v,a,s)3La2s+2f(t,x,v,a,s)23Las2+(3a24Lxv2s+6as4Lxvas+6af(t,x,v,a,s)4Lxvs2+3s24Lxa2s+6sf(t,x,v,a,s)4Lxas2+3f(t,x,v,a,s)24Lxs3+6a4Ltxvs+6s4Ltxas+6f(t,x,v,a,s)4Ltxs2+3a3Lx2s2a3Lxva+3s3Lxvs2s3Lxa2+f(t,x,v,a,s)3Lxas+34Lt2xs23Ltxa+2Lxv)v+3a4Lt2vs+3s4Lt2as+3f(t,x,v,a,s)4Lt2s2+3a3Ltxs2a3Ltva+3s3Ltvs2s3Lta2+f(t,x,v,a,s)3Ltasa2Lxa+s2Lxs+a2Lv2+(3s2+af(t,x,v,a,s))3Lvas+2f(t,x,v,a,s)2Lvsf(t,x,v,a,s)2La2+4Lt3s3Lt2a+2LtvLx\renewcommand{\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$^{nd}degree.These2 degree. These 2^{nd}degreetermscomefor degree terms come for N^2$ applied to Da(L)Da(L),

t0 = eq1c.expr().coefficient(f.expr()^2)*f.expr()^2; M.scalar_field(t0.expand()).display()
MR(t,x,v,a,s)3vf(t,x,v,a,s)24Lxs3+3af(t,x,v,a,s)24Lvs3+3sf(t,x,v,a,s)24Las3+3f(t,x,v,a,s)24Lts3+2f(t,x,v,a,s)23Las2\renewcommand{\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()
MR(t,x,v,a,s)vf(t,x,v,a,s)3Lxas+af(t,x,v,a,s)3Lvas+sf(t,x,v,a,s)3La2s+f(t,x,v,a,s)23Las2+f(t,x,v,a,s)3Ltas\renewcommand{\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()
MR(t,x,v,a,s)f(t,x,v,a,s)23Las2\renewcommand{\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
[t,x,v,a]\renewcommand{\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()
MR(t,x,v,a,s)sL1(t,x,v,a)+L0(t,x,v,a)\renewcommand{\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]
f(t,x,v,a,s)=v33(x)3L1(t,x,v,a)+a33(v)3L1(t,x,v,a)+2a2s3(v)2aL1(t,x,v,a)+as23v(a)2L1(t,x,v,a)+(3a3(x)2vL1(t,x,v,a)+2s3(x)2aL1(t,x,v,a)3(x)2aL0(t,x,v,a)+33t(x)2L1(t,x,v,a))v2a23(v)2aL0(t,x,v,a)2as3v(a)2L0(t,x,v,a)s23(a)3L0(t,x,v,a)+3a23t(v)2L1(t,x,v,a)+4as3tvaL1(t,x,v,a)+s23t(a)2L1(t,x,v,a)+3a22xvL1(t,x,v,a)+2as2xaL1(t,x,v,a)+4as2(v)2L1(t,x,v,a)+3s22vaL1(t,x,v,a)+(3a23x(v)2L1(t,x,v,a)+4as3xvaL1(t,x,v,a)+s23x(a)2L1(t,x,v,a)2a3xvaL0(t,x,v,a)2s3x(a)2L0(t,x,v,a)+6a3txvL1(t,x,v,a)+4s3txaL1(t,x,v,a)+3a2(x)2L1(t,x,v,a)+4s2xvL1(t,x,v,a)23txaL0(t,x,v,a)+2xvL0(t,x,v,a)+33(t)2xL1(t,x,v,a))v2a3tvaL0(t,x,v,a)2s3t(a)2L0(t,x,v,a)a2xaL0(t,x,v,a)+a2(v)2L0(t,x,v,a)+3a3(t)2vL1(t,x,v,a)+2s3(t)2aL1(t,x,v,a)+3a2txL1(t,x,v,a)+4s2tvL1(t,x,v,a)3(t)2aL0(t,x,v,a)+2tvL0(t,x,v,a)xL0(t,x,v,a)+3(t)3L1(t,x,v,a)v2xaL1(t,x,v,a)+a2vaL1(t,x,v,a)2(a)2L0(t,x,v,a)+2taL1(t,x,v,a)+2vL1(t,x,v,a)\renewcommand{\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()
MR(t,x,v,a,s)v44Lx3s+(3a4Lx2vs+3s4Lx2as+3f(t,x,v,a,s)4Lx2s2+33Lxs2fx+2Ls22fx2+34Ltx2s3Lx2a)v3+(3a24Lxv2s+6as4Lxvas+6af(t,x,v,a,s)4Lxvs2+3s24Lxa2s+6sf(t,x,v,a,s)4Lxas2+3f(t,x,v,a,s)24Lxs3+3a3Lvs2fx+3s3Las2fx+3f(t,x,v,a,s)3Ls3fx+6a4Ltxvs+6s4Ltxas+6f(t,x,v,a,s)4Ltxs2+3a3Lx2s2a3Lxva+3s3Lxvs2s3Lxa2+f(t,x,v,a,s)3Lxas+3(afv+sfa+f(t,x,v,a,s)fs+ft)3Lxs2+(2a2fxv+2s2fxa+2f(t,x,v,a,s)2fxs+fxfs+22ftx)2Ls2+33Lts2fx+34Lt2xs23Ltxa+2Lxv)v2+(a34Lv3s+3a2s4Lv2as+3a2f(t,x,v,a,s)4Lv2s2+3as24Lva2s+6asf(t,x,v,a,s)4Lvas2+3af(t,x,v,a,s)24Lvs3+s34La3s+3s2f(t,x,v,a,s)4La2s2+3sf(t,x,v,a,s)24Las3+f(t,x,v,a,s)34Ls4+3a24Ltv2s+6as4Ltvas+6af(t,x,v,a,s)4Ltvs2+3s24Lta2s+6sf(t,x,v,a,s)4Ltas2+3f(t,x,v,a,s)24Lts3+3a23Lxvs+3as3Lxas+3af(t,x,v,a,s)3Lxs2a23Lv2a+3as3Lv2s2as3Lva2s23La3+sf(t,x,v,a,s)3La2s+3a4Lt2vs+3s4Lt2as+3f(t,x,v,a,s)4Lt2s2+3a3Ltxs2a3Ltva+3s3Ltvs2s3Lta2+f(t,x,v,a,s)3Ltas+3(afv+sfa+f(t,x,v,a,s)fs+ft)3Lts2a2Lxa+s2Lxs+a2Lv2+(3s2+af(t,x,v,a,s))3Lvas+2f(t,x,v,a,s)2Lvs+3(a2fv+asfa+af(t,x,v,a,s)fs+sf(t,x,v,a,s)+aft)3Lvs2f(t,x,v,a,s)2La2+(3asfv+3s2fa+3sf(t,x,v,a,s)fs+2f(t,x,v,a,s)2+3sft)3Las2+(a22fv2+2as2fva+2af(t,x,v,a,s)2fvs+s22fa2+2sf(t,x,v,a,s)2fas+sfafs+f(t,x,v,a,s)fs2+f(t,x,v,a,s)22fs2+2a2ftv+2s2fta+2f(t,x,v,a,s)2fts+afx+(afs+s)fv+f(t,x,v,a,s)fa+ftfs+2ft2)2Ls2+3(af(t,x,v,a,s)fv+sf(t,x,v,a,s)fa+f(t,x,v,a,s)2fs+f(t,x,v,a,s)ft)3Ls3+4Lt3s3Lt2a+2LtvLx)v\renewcommand{\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 vv

v*(N(p)-Dx(L)) == (Dt(L)-(N(L)-(p*a+q*s+r*f)-(v*N(p)+a*N(q)+s*N(r))))
True\renewcommand{\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()
(vp(t,x,v,a,s)aq(t,x,v,a,s)sr(t,x,v,a,s)+L(t,x,v,a,s))dt+p(t,x,v,a,s)dx+q(t,x,v,a,s)dv+r(t,x,v,a,s)da\renewcommand{\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
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}
d(alpha).display()
(vpx+aqx+srxLx+pt)dtdx+(vpv+aqv+srv+p(t,x,v,a,s)Lv+qt)dtdv+(vpa+aqa+sra+q(t,x,v,a,s)La+rt)dtda+(vps+aqs+srs+r(t,x,v,a,s)Ls)dtds+(pv+qx)dxdv+(pa+rx)dxdapsdxds+(qa+rv)dvdaqsdvdsrsdads\renewcommand{\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)
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}
ev(alpha)(N)==L
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}
Omega = -(p*dx+q*dv+r*da).wedge(d(t)); Omega.display()
p(t,x,v,a,s)dtdx+q(t,x,v,a,s)dtdv+r(t,x,v,a,s)dtda\renewcommand{\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)
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}

Equation of Motion (E=0E=0)

E = ev(N)(d(alpha)) E.display()
(v2pxa2qvasqaaf(t,x,v,a,s)qsasrvs2rasf(t,x,v,a,s)rs(apv+spa+f(t,x,v,a,s)ps+aqx+srxLx+pt)vap(t,x,v,a,s)sq(t,x,v,a,s)f(t,x,v,a,s)r(t,x,v,a,s)+aLv+sLa+f(t,x,v,a,s)Lsaqtsrt)dt+(vpx+apv+spa+f(t,x,v,a,s)psLx+pt)dx+(vqx+aqv+sqa+f(t,x,v,a,s)qs+p(t,x,v,a,s)Lv+qt)dv+(vrx+arv+sra+f(t,x,v,a,s)rs+q(t,x,v,a,s)La+rt)da+(r(t,x,v,a,s)Ls)ds\renewcommand{\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)
True\renewcommand{\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)
True\renewcommand{\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
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}
d(p*v) == v*d(p) + p*d(v)
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}
d(q*a) == a*d(q) + q*d(a)
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}
d(r*s) == s*d(r) + r*d(s)
True\renewcommand{\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
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}
N(p*v) == p*N(v) + v*N(p)
True\renewcommand{\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
True\renewcommand{\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)
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}
d(L) == Dt(L)*dt + Dx(L)*dx + Dv(L)*dv + Da(L)*da + Ds(L)*ds
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}
r=Ds(L); r.display()
MR(t,x,v,a,s)Ls\renewcommand{\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()
MR(t,x,v,a,s)v2Lxsa2Lvss2Lasf(t,x,v,a,s)2Ls22Lts+La\renewcommand{\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()
MR(t,x,v,a,s)v23Lx2s+a23Lv2s+2as3Lvas+2af(t,x,v,a,s)3Lvs2+s23La2s+2sf(t,x,v,a,s)3Las2+f(t,x,v,a,s)23Ls3+(2a3Lxvs+2s3Lxas+2f(t,x,v,a,s)3Lxs2+2Ls2fx+23Ltxs2Lxa)v+2a3Ltvs+2s3Ltas+2f(t,x,v,a,s)3Lts2+a2Lxsa2Lva+s2Lvss2La2+(afv+sfa+f(t,x,v,a,s)fs+ft)2Ls2+3Lt2s2Lta+Lv\renewcommand{\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()
MR(t,x,v,a,s)4m2v68mv4V(x)+5a2222sv8mv4\renewcommand{\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()
MR(t,x,v,a,s)24mv3\renewcommand{\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()
MR(t,x,v,a,s)4m2v68mv4V(x)+5a228mv4\renewcommand{\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}}{8 \, m v^{4}} \end{array}
Lp == Lp0 + s*Lp1
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}
eq1p = eq1.expr().substitute_function(L.expr().operator(),Lp.expr().function(*list(coord))); eq1p
12a(6(5m2v44mv2V(x))mv44(12m2v516mv3V(x)2s)mv5+5(4m2v68mv4V(x)+5a2222sv)mv6)10a32mv6+a2smv5+2f(t,x,v,a,s)4mv4+xV(x)\renewcommand{\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())
[f(t,x,v,a,s)=2(2am2v6+2mv6xV(x)+5a324a2sv)2v2]\renewcommand{\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]