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The Hamiltonian is defined as the Legendre transform of the Lagrangian, with respect to a particular set of coordinates. The formula for that transform uses the chosen coordinates. So the Hamiltonian is not well-defined unless we can be certain that the value will be the same if we use any different set of coordinates to perform the Legendre transformation.

The derivations I have seen have not addressed this point. They just seem to assume the Hamiltonian will be well-defined.

Am I missing something obvious here? Is there a simple reason why the Hamiltonian's value will not depend on the coordinates used to derive it?