What are the problems with a path dependent potentials?

[Shing Ernst]

What exactly are the problems with a path-dependent potentials? (e.g. magnetic scalar potential, normal electric potential over a changing electric field)
I came up with this question, but can't be sure.
The problems with such potential seem to me may be multivalued, as well as making the computation too complicated.

 

[FactChecker]

Every little wiggle in the path can change the value. Also, returning to the original position will not get you back to the original state unless you exactly reverse the path. You are not dealing with a conservative force or a potential energy function.

 

[Shing Ernst]

I am thinking if there is actually nothing wrong with path dependent potentials? just because path-independent potential so powerful that some of us split on path dependent potentials...? (Since sometimes we can calculate things in seconds thanks to conservation of energy, which is guaranteed by path-independent potential)
(btw, a friend of mine on the Internet reminded me a few search on Google, it seems has something to do with Feynman's path integral.)

 

[FactChecker]

I am thinking if there is actually nothing wrong with path dependent potentials? just because path-independent potential so powerful that some of us split on path dependent potentials...? (Since sometimes we can calculate things in seconds thanks to conservation of energy, which is guaranteed by path-independent potential)
(btw, a friend of mine on the Internet reminded me a few search on Google, it seems has something to do with Feynman's path integral.)

If you are dealing with something that is path dependent, then you have no choice. But just try writing one down and you will see that it is difficult. And even if you do that, the things you can say about it only apply to that one path. Luckily, so much interesting and important physics deals with conservative forces and potentials (gravity, electro-magnetic, etc. ) that it is not much of a problem. And a lot more can be done if the paths are restricted to avoid circling around singularities.