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Can I always find a Hamiltonian for any given Dynamical System such that the Hamiltons' equations are satisfied? The hamiltonian may be an extremely complicated function (Possibly containing complex terms) but in principle, is it always possible to find the hamiltonian for a given Dynamical System?

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What is your definition of dynamical system? In general, the answer is no, because dynamical systems don't even have to involve derivatives! – Christopher A. Wong Mar 2 at 6:52
Even for continuous-time dynamical systems, the answer is no. Being Hamiltionan is a very special property for a system to have. – Hans Lundmark Mar 2 at 15:05
I would recommend you to answer yourself this important question by just proving that it is impossible for an autonomous Hamiltonian system (independent of time) to be asymptotically stable. In my opinion, it is a nice and very enlightening excercise. This implies that those Hamiltonian systems conserve energy, there is no dissipation of it, so for example a mathematical damped pendulum could not be described by a Hamiltonian system. For a very quick review of this things you could check scholarpedia.org/article/Hamiltonian_systems – user58533 Mar 14 at 10:48

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In classical dynamics you can apply Hamilton's Principle to every dynamical system, because you can work with generalized coordinates. According to E.B. Wilson (MIT) Hamilton's Principle, which contains the Principle of Least Action as a special case, is the most fundemantal and important single theorem in mathematical physics.

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