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I was wondering whether it is correct to say that if two components of the angular momentum are conserved, then all three Cartesian coordinates of the angular momentum are conserved? I would regard this as a direct consequence from the Poisson bracket. Is this correct so far?

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up vote 7 down vote accepted

Yes, because $\{J_i,J_j\}_{PB}=\sum_{k=1}^3\epsilon_{ijk} J_k$. More generally, the statement that the Poisson bracket of any two constants of motion is again a constant of motion is known as Poisson's Theorem.

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