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?

  • $\begingroup$ What ideas do you have about this? $\endgroup$ Feb 14 '17 at 6:02
  • $\begingroup$ never mind .... $\endgroup$ Feb 14 '17 at 6:19
  • $\begingroup$ 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? $\endgroup$ Feb 14 '17 at 6:33
  • $\begingroup$ @sammygerbil, it was admittedly just a thought - The inspiration was because I recently learnt that a brachistochrone curve (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? $\endgroup$ Feb 14 '17 at 6:47
  • $\begingroup$ Looks like I'm wrong - Motion on the brachistochrone curve is indeed SHM. $\endgroup$ Feb 14 '17 at 7:00

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