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Classical Mechanics, by John Taylor defines a conservative force F as a force that satisfies:

  1. F depends only on the particle's position and no other variables.

  2. 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)?

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  • $\begingroup$ What if the force is dependent only on the particle's position and on time? $\endgroup$ – probably_someone Feb 11 '18 at 20:23
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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).

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