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?