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If my thinking is wrong please let me know. I have little knowledge on beyond-college physics. For research purposes, I read a few introductions to these three formalisms of classical mechanics [1,2,...

I know what is the Liouville integrability: given a Hamiltonian with $n$ degrees of freedom, with $n$ independent constants of motion in involution, the Hamiltonian can be brought to the form $H(p_1, \...