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?
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$\begingroup$ en.wikipedia.org/wiki/Hamiltonian_mechanics#Overview $\endgroup$– SuperCiociaCommented Oct 1, 2017 at 1:51
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$\begingroup$ Possible duplicates: physics.stackexchange.com/q/51510/2451 , physics.stackexchange.com/q/282724/2451 . Related: physics.stackexchange.com/q/56243/2451 , physics.stackexchange.com/q/175021/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Oct 1, 2017 at 4:45
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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 $\dot{q} = p / m,\ \dot{p} = -dV(q)/dq$.
