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Is there another type of motion other than SHM that has this property? How would one systematically find the general form of the motion that respects this constraint?

user1936752
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  • What ideas do you have about this? – sammy gerbil Feb 14 '17 at 06:02
  • never mind .... – ZeroTheHero Feb 14 '17 at 06:19
  • There are a lot of open questions here: do you allow for time dependent forces? Do you allow non conservative forces or forces that depend on the nth time derivative, like e.g. friction? – mikuszefski Feb 14 '17 at 06:33
  • @sammygerbil, it was admittedly just a thought - The inspiration was because I recently learnt that a brachistochrone curve (https://en.wikipedia.org/wiki/Brachistochrone_curve) obeys this property and the motion of a frictionless bead on such a curve is not simple harmonic.

    Assuming you allow a force that depends on the displacement and its time derivatives, is the question still too broad?

     – user1936752 Feb 14 '17 at 06:47
  • Looks like I'm wrong - Motion on the brachistochrone curve is indeed SHM. – user1936752 Feb 14 '17 at 07:00
  • I don't understand what the question is. This is true for any linear system whose response happens to be periodic. SHM is just a very special (and simple) case. – alephzero Feb 14 '17 at 16:51

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