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Suppose a force F = F(r, t) where r is a three dimensional space vector and t is time.

I understand that in order to a force be conservative two conditions must be satisfied:

  1. The force must be distance dependent only, i.e. F = F(r).
  2. The work done by such force between two points, say, 1 and 2, must be independent of the path taken.

Firstly, we may automatically say that the force F that I am talking about is not conservative since it has time dependence.

However, it can be a force such that its curl is 0, i.e. $\nabla \times \textbf{F} = 0$. If that's the case, it's sufficient to argue that this force is such that the work done by this force between two points is path invariant.

I wonder whether F may be derived from a potential energy even though it does not satisfay condition 1, but satisfy condition 2.

All of this being said, I want to ask: One is always able to derive a force from a potential energy or is there any conditions to do so? It must satisfy any of the conditions mentioned above? It must satisfy both (be conservative)? Even further, if not in the force, is there any conditions that the potential energy must satisfy in order to this relation hold?

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The problem with a Force that depends on time and not just on position, is that regarding time, you can always change the work in a path because you can wait until the force changes

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