I know Hamiltonian can be energy and be a constant of motion if and only if:
- Lagrangian be time-independent,
- potential be independent of velocity,
- coordinate be time independent.
Otherwise $$H\neq E\neq {\rm const},$$ or $$H=E\neq {\rm const},$$ or $$H\neq E={\rm const}.$$
I am looking for examples of these three situation.
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