Question

When is the Hamiltonian of a system not equal to its total energy
isnt the two the same thing?!

Answer 1

I mean H=T=V whicg is defination of energy, right?

Answer 2

sorry, T+V=H=total E, so what can make the second equal sign fail?

Answer 3

maybe ... the hamiltonian, though, works with momentum, as far as I forget Hamiltonian mechanics
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Answer 4

I don't know if the derivation of Hamiltonian mechanics is something you've encountered in class, but the way to get to Hamiltonian mechanics is to apply the Legendre transformation. The transformation boils down to:
$$L(q_j, \dot q_j, t) \rightarrow H\left(q_j, \frac {\partial \mathbf L}{\partial \dot q_j }, t\right)$$
Which implies that
$$\frac {\partial \mathbf H}{\partial t} = - \frac {\partial \mathbf L}{\partial t}$$
So for **neither** the Lagrangian or the Hamiltonian of a conservative (or in general a potential) system do not depend on time explicitly.

This constraint is known as holonomic constraint, and only when the holonomic constraint is fulfilled the Hamiltonian will represent the total energy of the system. This of course doesn't mean you can't write a Hamiltonian in nonholonomic system, just that it won't directly represent the total energy of the system.

In a more simple language, if your generalized coordinates explicitly depend on time you're dealing with a nonholonomic system. Then Hamiltonian isn't total energy of the system. So all accelerating or moving frames of reference are out of the question.

At best you can use Noether theorem to prove that the quantities that (anti?)commutate with Hamiltonian (or action of the system) will be conserved but that's beyond what I can remember exactly...