(BASIC) Deriving Hamiltonian from equations of motion

Question

Suppose that some particle can be described by a system of differential equations with respect to displacement and time. Is there a general procedure of extracting the hamiltonian of such a system?

Answer 1

Nope. Many systems' equations of motion are not equivalent to Hamilton's equations for any Hamiltonian, so there is no Hamiltonian formulation for such systems. Search "non-Hamiltonian system" for examples. Such systems usually include friction, dissipation, or something else that violates conservation of energy. (However, some systems with friction or dissipation can still be described by a Hamiltonian that depends explicitly on time, or has position-momentum cross terms - e.g. http://www.hep.princeton.edu/~mcdonald/examples/damped.pdf.)

Even if you do somehow know that your equations of motion do correspond to some Hamiltonian, I do not believe that there's any known general procedure for reconstructing that Hamiltonian, unless of course your equations of motion are simple, like $\dot{q} = p / m,\ \dot{p} = -dV(q)/dq$.