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We know that 'Hamilton's equations' preserve in canonical transformation. But does this mean than 'Hamiltonian' itself doesn't change under canonical transformation?

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No, the Hamiltonian can transform $H\to K = H +\frac{\partial F}{\partial t}$ under a CTs that have a generator $F$ with explicit time dependence.

References:

  1. H. Goldstein, Classical Mechanics; Chapter 9.
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