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I am currently taking a course in theoretical (classical) Mechanics, where I have learned about the Darboux theorem. My professor has also mentioned one can "reduce the system by symmetry", meaning that if we find a conserved quantity, we can separate the phase space $T^*\mathbb Q$ in a sub-manifold of dimension 2 and another one of dimension $2n-2$.

Therefore, if one can find $n$ conserved quantities (and which are in involution, that is, they all commute in the Poisson sense) then we can completely separate the symplectic manifold in $n$ sub-phase spaces of one degree of freedom each (hence, 2-dimensional).

My question is whether we can use these conserved quantities as our new variables in order to describe the dynamics of the system. For instance, for an object orbiting Earth, would it be equivalent to provide $(E, \vec L^2, L_z)$ than to provide the trajectory of the object?

This question is rather obviously influenced by Quantum Mechanics, where we precisely use these conserved quantities to label states of a system. Therefore, I wonder whethere we can use conserved quantities to label the mechanical state of a classical system, too.

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  • $\begingroup$ The answer to this is trivially yes, and you are meant to do that. I do not even know of a textbook discussing Hamiltonian mechanics that fails to mention this. Goldstein certainly does mention this, and even discusses the Roothian. Goldstein also takes pain to describe in very high detail the Hamilton–Jacobi formulation that reduces problem-solving into two different beautifully nice solution forms. $\endgroup$ Commented Nov 26 at 20:26
  • $\begingroup$ What do you mean use them as variables. They are constant, they don’t vary. What is it you are looking to do with them? If you just want to know the shape of the orbit then they are as good as the whole trajectory (ellipse), if you want to know the trajectory in the sense of knowing where the particle as a function of time, that’s not going to work. $\endgroup$ Commented Nov 26 at 21:20
  • $\begingroup$ Actually E,L^2,Lz are NOT equivalent to the trajectory. The trajectory has six degrees of freedom and you only have three. If you had E, the L vector, and the Runge-Lentz vector, that would do it. Or if you want to go old school semi-major axis, eccentricity, inclination, right ascension of the ascending node, argument of periapsis, and true anomaly. I guess it’s not clear what you are really asking. $\endgroup$ Commented Nov 26 at 21:31

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For what it's worth, it seems relevant in OP's context of a coompletely Liouville integrable system to mention the notion of angle-action coordinates $(w^1,\ldots, w^n,J_1,\ldots, J_n)$, and the Liouville-Arnold theorem, cf. e.g. this Phys.SE post. Here the action variables $(J_1,\ldots, J_n)$ are $n$ Poisson commuting conserved quantities.

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