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How can I prove that every one-dimensional system is integrable (meaning that there is a constant of motion)?

It is clear that if $H$ does not depend explicitly on time then $H$ is indeed a constant of motion, (that is the energy).

If $H$ is time dependent I have tried various examples but found no common constant. Is there a known way to find such constants of motion? I am aware that those constants may be related to Hamilton-Jacobi equations or Louiville's Theorem. Could you please indicate how to find such a constant in general.

Qmechanic
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SultanDeGranada
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    Hi SultanDeGranada Welcome to Phys.SE. Do you mean a globally defined constant of motion? Who makes the claim for 1D systems with explicit time-dependence? Which page? – Qmechanic Jun 18 '22 at 10:00
  • At a Lecture we were asked to prove it. – SultanDeGranada Jun 18 '22 at 10:05
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    And the lecturer specifically mentioned explicit time-dependence? – Qmechanic Jun 18 '22 at 10:11
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    If the system is "autonomous" (no explicit time dependence, https://en.wikipedia.org/wiki/Autonomous_system_(mathematics) ), see this answer and links therein: https://physics.stackexchange.com/a/583869/226902 . More useful terminology here: https://physics.stackexchange.com/q/55861/226902 – Quillo Jun 18 '22 at 10:30
  • He did not made explicit mention of time dependece but otherwise it is trivial. – SultanDeGranada Jun 18 '22 at 10:51

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