Graduate Test if 2nd order diff eq. can be derived from a Hamiltonian

[Bosh]

Imagine I have a complicated second-order differential equation that I strongly suspect can be derived from a Hamiltonian (with additional momentum dependence beyond p²/2m, so the momentum is not simply mv, but I don't know what it is).

Are there any ways to test whether or not the given second order differential equation can be derived from a Hamiltonian without finding what it is?

 

[hilbert2]

If the potential energy of the system doesn't have an infinity (like in the potential of two point charges: $V(r)\propto r^{-1}$), you can try to form a general Lagrangian function

$L =\sum_{k,l=0}^\infty c(k,l)\dot{q}^{k}q^{l}$

where the $c(k,l)$ are constant coefficients, and calculate the corresponding eq. of motion and find by comparison what Lagrangian corresponds to you DE. However, if there's fractional powers of x and p in the equation, this probably isn't so easy.

 

[Bosh]

This has been my approach. I should have added that the additional terms are a perturbation, so I'm trying to achieve even just the simpler goal of matching terms to the next order in epsilon. Even so, I'm only close in a simple case, and in complicated cases it seems hopeless to get everything to match up (it's not a 1 degree of freedom problem, and there are many terms to simultaneously match).

So I was trying to think if there's a more clever, non-brute-force method. Something like trying to numerically integrate the equations of motion and somehow checking the dimensionality of the manifold on which the trajectories move to check for conserved quantities. I hoped though that there might be something simpler...like connecting the differential equation to some sort of canonical transformation and checking whether a symplectic criterion is satisfied...

Thanks for any thoughts!