vanhees71 said:Well, the section
https://en.wikipedia.org/wiki/Conservative_force#Mathematical_description
is inaccurate (mathematically spoken it's simply wrong). The Wikipedia author(s) forgot that the Poincare lemma is valid only in simply-connected regions of space. The most famous counter example is the potential vortex. See here:
https://www.physicsforums.com/threads/struggling-with-ab-effect.872156/#post-5477281
The formal definition of a conservative force is a force that does not depend on the path taken by an object, but only on its initial and final positions. This means that the work done by a conservative force is independent of the path taken by the object.
Some common examples of conservative forces include gravity, electric forces, and magnetic forces. These forces are conservative because they do not depend on the path taken by an object, but only on its initial and final positions.
A conservative force is related to potential energy through the work-energy theorem, which states that the work done by a conservative force is equal to the change in potential energy of an object. This means that as an object moves through a conservative force field, its potential energy changes in a predictable way.
The significance of a conservative force in physics is that it allows us to simplify and analyze systems using the principle of conservation of energy. Conservative forces also play a crucial role in understanding and predicting the behavior of objects in nature.
A force is conservative if it meets two criteria: 1) the work done by the force is independent of the path taken by an object, and 2) the work done by the force is equal to the negative change in potential energy. If both of these criteria are met, then the force is considered conservative.