# Conservation of Hamiltonian vs Conservation of Energy

What is the difference between conservation of the Hamiltonian and conservation of energy?

-

$$\frac{dH(q,p,t)}{dt}=\frac{\partial H}{\partial q}\dot{q}+\frac{\partial H}{\partial p}\dot{p}+\frac{\partial H}{\partial t}=-\dot{p}\dot{q}+\dot{q}\dot{p}+\frac{\partial H}{\partial t}$$
From this you see that the Hamiltonian is conserved if it does not depend on time,$t$, explicitly. $H$ may or may not be the total energy, if it is, this means the energy is conserved. But even if it isn't, $H$ is still a constant of motion.
The potential is velocity independent in order to have conservative forces. If $V=V(q,\dot{q})$, then the forces wouldn't be conservative and you couldn't state that $H=E$ –  vnb Nov 28 '13 at 15:06