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.