Periodic Orbits: The Role of the Coulomb Potential in Producing Periodic Motion

[lalbatros]

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

 

[pervect]

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.

 

[lalbatros]

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]

I've never had any deeper insight than the mathematical demonstration you've already seen, sorry.