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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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.