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A typical description of a central force is found here Lecture L15 - Central Force Motion: Kepler’s Laws (pdf).

"When the only force acting on a particle is always directed to­ wards a fixed point, the motion is called central force motion."

The two cases I am familiar with are the inverse square force, and the mass on a spring.

I am assuming a free test particle (e.g., a planet) not subject to friction nor any other force.

Other that having a "central attractor", what other conditions must be met to qualify as a "central force problem"? For example, can the force at a field point change with time? Can the force depend on radial angle? This latter proposition appears to require a potential gradient with a non-radial component, so violates the central force requirement.

Qmechanic
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Steven Thomas Hatton
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1 Answers1

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Taken literally, a central force problem is one in which the force on a particle is given by $$\mathbf F(\mathbf r,t) = f(\mathbf r,t) \hat r$$ where $\hat r \equiv \mathbf r/|\mathbf r|$. This is a rather broad class of physical models, and not much can be said about them at this level of generality. Most of the time when we talk about central forces, we mean forces which can be derived from a time-independent central potential, i.e. $$\mathbf F(\mathbf r,t) = -\nabla V(\mathbf r)$$ These are the forces which e.g. conserve angular momentum, and it is these forces which are the subject of Bertrand's theorem on stable orbits.

As far as I know, there is no universal convention as to which type is meant when one says the words "central force problem." Presumably if you come across a discussion of central forces, a reasonably careful author would define what those words are supposed to mean. If they don't, then it will probably become apparent in short order which definition they're using, and if it doesn't, then that would be a good time to ask.

This latter proposition appears to require a potential gradient with a non-radial component, so violates the central force requirement.

This argument fails because a generic force simply can't be derived from a potential. If the force is conservative, then this argument holds, which is a quick way to see that conservative central forces must be spherically symmetric (i.e. $V(\mathbf r) = V(r)$).

Albatross
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