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Someone told me that, in a hamiltonian system, the hamilonian function is the generating function of the canonical transformation given by time translation. However, this statement doesn't make any sense to me. Typically, the generating function is a function of some of the "old" coordinates as well as the new (transformed) ones, but the hamiltonian is a function of q and p.

Please tell me whether the above statement is rubbish, or whether it can indeed be interpreted in a meaningful way.

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up vote 5 down vote accepted

The statement isn't that $H$ is a generating function of the canonical transformation, in the sense used in canonical transformations, which must indeed be a function of one new and one old set of phase space coordinates. The statement is that $\hat H$ is the infinitesimal generator of the Lie group of translations in the sense of the theory of Lie algebras, namely a basis vector of the Lie algebra. These are different words – generating function and generator. Both of them "generate" the transformations of the phase space variables in some way but the two ways are technically different. In classical physics, the infinitesimal generator generates infinitesimal canonical transformations via the Poisson bracket (the classical limit of the commutator of linear operators), $$ q_i \to q_i+\delta q_i,\quad \delta q_i = \epsilon\{ H,q_i \} $$ where $q_i$ may be both coordinates or momenta.

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