Steven W. answered • 07/31/19

Physics Ph.D., college instructor (calc- and algebra-based)

There are three mathematical ways of describing a conservative force, one of which you list above. There are two others, which are equivalent IF the force in question is a field force (in other words, can be represented as a force field).

An essential part of being a force field is that, once the environment is set, the force at any point depends only on the location of that point. No matter how a particle may be moving through that point, the force on it there depends only on being at that position.

The magnetic force, however, cannot be represented this way, since the magnitude and direction of the magnetic force on a (charged) particle depends not only on where the particle is, but also how fast it is moving and which way. Two particles of identical size and electric charge, moving through the same position, may have a different magnetic force (magnitude and direction), if their speeds and directions different.

Since the magnetic force is not a force field, satisfying only one of the mathematical definitions of conservative is insufficient. And it turns out the magnetic force does not satisfy either of the other two, so it is officially NOT conservative, even though it may share some attributes with conservative forces.