I Potentials of conservative forces

AI Thread Summary
The discussion centers on the definition of conservative forces and potentials, specifically questioning whether potentials of the form V(r, \dot{r}) can be considered conservative. It is noted that a conservative force is defined by having a closed loop integral equal to zero, which suggests that V(r, \dot{r}) would not qualify as conservative. The conversation also touches on the implications of Goldstein's work and the relationship between time independence in Lagrangian mechanics and the conservation of energy. Additionally, there is a semantic debate regarding the classification of V(𝑥, 𝑑𝑥) as a potential. The thread emphasizes the importance of understanding the mathematical framework behind conservative forces in physics.
Kashmir
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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.
 
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PeroK said:
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.
 
Kashmir said:
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!
 
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
$$p_k=\frac{\partial L}{\partial \dot{q}^k}.$$
 
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