Classical Mechanics, by John Taylor defines a conservative force $F$ as a force that satisfies:
$F$ depends only on the particle's position and no other variables.
Work done by $F$ is the same for all paths taken between two points
I'm wondering if this definition is redundant. Doesn't (1) imply (2) and vice versa?
If not, what is an example of a force that satisfies (1) but not (2) and an example of a force that satisfies (2) but not (1)?
The comment of @probably_someone shows clearly the necessity of (1). It eliminates a possible force dependence on time, velocity or on any other parameters.
(2) does not follow from (1): Consider the force on one pole of a long thin bar magnet which is next to a current carrying wire. The work done moving it in a circle around the wire is different to the work done in a loop which doesn't go around the wire. The same would be the location dependent force on an object moved in a water whirl.
(1) doesn't follow from (2): When a charged particle moves in a magnetic field no work is done on the particle on going on any path from A to B. The force experienced by the particle is dependent on the velocity not only the position (inhomogeneous B).
