Question about the formal definition of conservative force

[physics user1]

So, I was reading the mathematical description of a conservative force o wikipedia : https://en.m.wikipedia.org/wiki/Conservative_force and at the line "Many forces (particularly those that depend on velocity) are not force fields. In these cases, the above three conditions are not mathematically equivalent".
Why these statements are not mathematically equivalent?

 

[vanhees71]

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

 

[physics user1]

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

Thanks!