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In wikipedia, in the page for constant of motion, it says

"In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion."

And suggests "methods for identifying constants of motion" and among them there is Noether's theorem. And in wikipedia for Noether's theorem, it also says

"The conservation law of a physical quantity is usually expressed as a continuity equation."

I was wondering, if I want to prove that a quantity is constant of motion, does it make sense to prove that the equation of continuity is zero, which shows that the time derivative of that quantitative is zero, which is as if "poisson bracket with the Hamiltonian equals minus its partial derivative with respect to time"? Are total derivatives for each case are different?

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  • $\begingroup$ A continuity equation is one (but not the only) way to show that a quantity is a constant of motion. $\endgroup$ – Qmechanic Nov 22 '19 at 12:08
  • $\begingroup$ Mathematically speaking there is a difference between Poisson's theorem and writing continuity equation. Although, first term is same, partial time derivate of the quantity, second term created some ambiguity for me,since applying Poisson's theorem to the quantity and taking divergence of the same quantity is different. $\endgroup$ – Jalhik Nov 22 '19 at 12:33
  • $\begingroup$ The last paragraph (v3) is a bit unclear. It sounds like OP is essentially asking this question. $\endgroup$ – Qmechanic Nov 22 '19 at 12:37

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