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Periodic orbits

  1. Oct 29, 2006 #1
    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?


  2. jcsd
  3. Oct 29, 2006 #2


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    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.
  4. Oct 29, 2006 #3
    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.


    PS: In other words, can a property of an integral be understood in another way?
  5. Oct 30, 2006 #4


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