%typeset_mode True
from sage.manifolds.utilities import exterior_derivative as d
def ev(N): return (lambda x: N.contract(x))
M = Manifold(5,'M')
coord.<t, x, v, a, s> = M.chart()
Vectors (partial derivatives)
[Dt,Dx,Dv,Da,Ds] = coord.frame()
[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())
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()
Similarly the first-order terms arise from $N$ applied to the term $Ds\ f$ in $N(Da(L))$.
N(f).display()
t1 = eq1a.expr().coefficient(diff(f.expr(),t))*diff(f.expr(),t); M.scalar_field(t1).display()
x1 = eq1a.expr().coefficient(diff(f.expr(),x))*diff(f.expr(),x); M.scalar_field(x1).display()
v1 = eq1a.expr().coefficient(diff(f.expr(),v))*diff(f.expr(),v); M.scalar_field(v1).display()
a1 = eq1a.expr().coefficient(diff(f.expr(),a))*diff(f.expr(),a); M.scalar_field(a1).display()
s1 = eq1a.expr().coefficient(diff(f.expr(),s))*diff(f.expr(),s); M.scalar_field(s1).display()
Liebniz rule applies $N$ to these coefficients.
c1 = 3*N(Ds(Ds(L)))*N(f); c1.display()
bool(c1.expr()==t1+x1+v1+a1+s1)
Removing the 1$^\textit{st}$ order terms:
eq1b = eq1a-c1; eq1b.display()
Only terms algebraic in $f$ remain. The number of terms is:
len(eq1b.expr().expand())
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()
cf3 = Ds(Ds(Ds(Ds(L))))*f^3; cf3.display()
Removing the 3$^{rd}$ degree term:
eq1c = eq1b-cf3; eq1c.display()
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()
cf2 = N(Ds(Da(L)))*f; cf2.display()
cf2 = Ds(Ds(Da(L)))*f^2; cf2.display()
We cannot solve this for f
algebraically since f
appears as derivatives.
Lagrangian linear in s.
ll=list(coord);ll.remove(s);ll
Ll = M.scalar_field(function('L0')(*ll)) + s * M.scalar_field(function('L1')(*ll)); Ll.display()
solve(eq1.expr().substitute_function(L.expr().operator(),Ll.expr().function(*list(coord))),f.expr())[0]
Second Equation of Lagrange
(Dt(L)-(N(L)-(p*a+q*s+r*f)-(v*N(p)+a*N(q)+s*N(r)))).display()
v*(N(p)-Dx(L)) == (Dt(L)-(N(L)-(p*a+q*s+r*f)-(v*N(p)+a*N(q)+s*N(r))))
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)
alpha = L*dt + p*(dx-v*dt) + q*(dv-a*dt) + r*(da-s*dt)
alpha.display()
alpha == L*dt+p*dx+q*dv+r*da-(p*v+q*a+r*s)*dt
d(alpha).display()
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)
ev(alpha)(N)==L
Omega = -(p*dx+q*dv+r*da).wedge(d(t)); Omega.display()
alpha == L*dt + ev(N)(Omega)
Equation of Motion ($E=0$)
E = ev(N)(d(alpha))
E.display()
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)
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)
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
d(p*v) == v*d(p) + p*d(v)
d(q*a) == a*d(q) + q*d(a)
d(r*s) == s*d(r) + r*d(s)
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
N(p*v) == p*N(v) + v*N(p)
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
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)
d(L) == Dt(L)*dt + Dx(L)*dx + Dv(L)*dv + Da(L)*da + Ds(L)*ds
r=Ds(L); r.display()
q=Da(L)-N(r); q.display()
p=Dv(L)-N(q); p.display()
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()
Lp1 = Ds(Lp); Lp1.display()
Lp0 = Lp - s * Lp1; Lp0.display()
Lp == Lp0 + s*Lp1
eq1p = eq1.expr().substitute_function(L.expr().operator(),Lp.expr().function(*list(coord))); eq1p
solve(eq1p,f.expr())