What are the problems with a path dependent potentials?

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Discussion Overview

The discussion revolves around the challenges and implications of path-dependent potentials in physics, such as magnetic scalar potential and electric potential in changing electric fields. Participants explore the theoretical and practical aspects of these potentials, including their complexity and multivalued nature.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Some participants suggest that path-dependent potentials may be multivalued and complicate computations.
  • One participant notes that any small change in the path can alter the potential value, and returning to the original position does not guarantee returning to the original state unless the path is exactly reversed.
  • Another participant questions whether there is anything inherently wrong with path-dependent potentials, suggesting that the power of path-independent potentials may lead to a bias against the former.
  • A participant mentions that while path-dependent potentials can be challenging to work with, they are sometimes necessary, and interesting physics often involves conservative forces and potentials.
  • It is noted that when dealing with path-dependent scenarios, the analysis is limited to the specific path taken, which can restrict the applicability of conclusions drawn.

Areas of Agreement / Disagreement

Participants express differing views on the nature and implications of path-dependent potentials. While some highlight the difficulties and limitations, others argue that these potentials are not inherently problematic, indicating an unresolved debate.

Contextual Notes

Participants acknowledge that the complexity of path-dependent potentials may arise from their multivalued nature and the necessity of considering specific paths, which can limit generalizability.

Shing Ernst
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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.
 
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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.
 
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.)
 
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Shing Ernst said:
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.
 
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