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I'm struggling to understand the argument on p. 13 in Landau and Lifshitz that for a system with $N$ degrees of freedom there must be $2N-1$ integrals of motion.

In particular, I can't understand how this works for a free particle. Clearly, the system is translationally and rotationally invariant. I think that the angular momentum is independent of the linear momentum. So then it seems like there are 6 independent integrals of motion, one for each component of linear momentum, and one for each component of angular momentum. Where does this argument go wrong?

Any help is much appreciated.

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The resolution is that the 3 linear momenta $p_i$ and the 3 angular momenta $L_i$ are not independent integrals of motion. They satisfy a quadratic relation $\vec{p}\cdot \vec{L}=0$. So the 3D free particle has only 5 independent integrals of motion.

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