Integrals of motion for a free particle

Question

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.

Answer 1

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.