homework and exercises - The relation between Hamiltonian and Energy

I know Hamiltonian can be energy and be a constant of motion if and only if:

1. Lagrangian be time-independent,


2. potential be independent of velocity,


3. 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.