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In Hamiltonian mechanics, consider extended phase space, the trajectory followed by a particle in that space is formed by an intersection of different 2n dimensional surfaces, all of these surfaces are constants of motion, are any symmetric transformations associated with these constants of motion?

what is the difference between constants of motion and conserved quantities formed from symmetric properties?

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Concerning OP's title question (v2): A constant of motion and a conserved quantity are the same notion.

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  • $\begingroup$ so for a system in extended phase space there are 2n constants of motion implies 2n conserved quantities does this implies 2n symmetric transformations can be done in that extended phase space??????? $\endgroup$ – robin raj Oct 11 '18 at 14:48
  • $\begingroup$ Yes, see e.g. this Phys.SE post. $\endgroup$ – Qmechanic Oct 11 '18 at 14:53

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