# Periodic orbits

Let's consider the motion of a test particle in a central field.
Is the Coulomb potential, 1/r, the only one that produces a periodic motion?
If no, is there a condition for periodicity to occur?

Thanks,

Michel

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pervect
Staff Emeritus
Most potentials will have periodic oribts, but many of them won't be closed periodic orbits. You can use the idea of an "effective potential" to make the problem one-dimensional, the period of the orbit would then be the time interval between apocenteron or pericentron. But if the angle covered wasn't 2*pi radians, the orbit will be periodic, but not closed.

There's a list in Goldstein "Classical Mechanics" of the force-laws that give closed orbits, IIRC.

Thanks a lot for your useful comment.

You gave me the idea to check in Landau-Lifchitz. (I don't have Goldstein unfortunately).
He states that there are only two potentials that result in closed trajectories: 1/r and r² . That's already good to know. However, I don't see where this magic comes from. The algebra is simple and clear, but it does not indicate some more "fundamental" reason.

Michel

PS: In other words, can a property of an integral be understood in another way?

pervect
Staff Emeritus