Question: Why is energy not conserved when the potential is dependent on velocity? If not, then why not?
Attempt: I know that if the potential is dependent on time, then energy is not conserved. A velocity $v$ can also be expressed as $v=x/t$, from where an explicit time dependents arises, but I'm not so sure if this is the right way.
edit clarity: The Hamiltonian is not the total energy if $V=V(\mathbf{q},\mathbf{\dot{q}})$ but can still be conserved from my understanding. Here I mean the total energy $E=T+V$. Is $E$ still conserved even when $V=V(\mathbf{q},\mathbf{\dot{q}})$? I think I struggle a bit with differentiating in which cases $H$ is the total energy and/or conserved $and$ $E=T+V$ is the total energy (which should always be the case right?) and/or conserved.
