Potentials of conservative forces

[Kashmir]

Goldstein writes

"only if $V$ is not an explicit function of time is the system conservative"That means $V(r,\dot{r})$ is a conservative potential, however I think that only potentials of the form $V(r)$ are conservative potentials.

Could you please tell me where I'm going wrong.

Thank you.

 

[PeroK]

There's a discussion here about whether the magnetic force is conservative or not.

https://en.wikipedia.org/wiki/Conservative_force

And, there is plenty of further discussion online.

 

[Kashmir]

There's a discussion here about whether the magnetic force is conservative or not.

https://en.wikipedia.org/wiki/Conservative_force

And, there is plenty of further discussion online.

If we define a conservative force those whose closed loop integral is zero, then $V(r,\dot{r})$ isn't conservative?
That's the definition the author began with.

 

[PeroK]

If we define a conservative force those whose closed loop integral is zero, then $V(r,\dot{r})$ isn't conservative?
That's the definition the author began with.

I'm not familiar with Goldstein, so I'm not sure what he's up to!

 

[vanhees71]

I'd not call $V(\vec{x},\dot{\vec{x}})$ a potential, but that's only a semantic question.

The math simply is that the first variation of the action is invariant under time translations if the Lagrangian is not explicitly time dependent, and then the Hamilotonian,
$$H=\dot{q}^k p_k - L=\text{const}$$
along the solutions of the Euler-Lagrange equations of motion, where the canonical momenta are defined by
$$p_k=\frac{\partial L}{\partial \dot{q}^k}.$$