In mathematics, the inverse problem for Lagrangian mechanics (Helmholtz inverse problem) is the problem of determining whether a given system of ordinary differential equations can arise as the Euler–Lagrange equations for some Lagrangian function.
For more information read section IV.2. page 65 of the following reference:
I need some hints or procedures (if it is possible) for similar (but a little more complex) problem:
1- Assume that you have one ordinary differential equation, ode1(r) in polar coordinate system (i.e. (r, theta)). The ODE is taken to be independent from theta (It is not a PDE).
2- Assume that "Euler" is an operator that gives the Euler-Lagrange equation, I need a procedure to calculate ode2(r) such that
1/(2r)*Euler (ode2(r)) -Laplacian (1/(2r)*Euler(ode1(r)))=0
It is obvious that we need inverse of Euler operator (say IE) to calculate ode2(r).
ode2(r) =IE( 2r*Laplacian (1/(2r)*Euler(ode1(r))))
I calculate ode2(r) for some simpler cases via trial and error method.
s := proc (S)
subs(w = w(r), w1 = diff(w(r), r), w2 = diff(w(r), r$2), S)
Euler := proc (f)
s(diff(f, w))-(diff(s(diff(f, w1)), r))+diff(s(diff(f, w2)), r$2)
ode1(r) = -r*(diff(w(r),r))^2:
ode2(r) = (diff(w(r),r))^2/r+r*(diff(w(r),r$2))^2:
I will be grateful if you can hint me to write an appropriate procedure.