Can any one please tell me in what cases the hamiltonian is not Equal to Total energy. My guess, albeit educated, is if the potential is either a function of time explicitly or a function of velocity, then $H\neq T_{energy}$.

Consequently, then, in every other case $H=T_{energy}$. Am I right? I would also appreciate of you would give me an example of a systems in which $H\neq T_{energy}$.

If I am wrong, then please correct me.