1. I was reading Landau & Lifschitz's book on Mechanics, and came across this sentence on p.19:
"There are no other additive integrals of the motion. Thus every closed system has seven such integrals: energy, three components of momentum, and three components of angular momentum".
However, no proof is given for this statement. Why is it true?
2. I find the statement somewhat counter-intuitive; it says at the beginning of the second chapter that for any mechanical system with $s$ degrees of freedom, there are at most $2s-1$ integrals of motion.
But the above statement would seem to imply that a system with three degrees of freedom has at least $2s+1$ integrals of motion. Why is this not a contradiction?
3. Finally, these integrals of motion correspond neatly to homogeneity of time (energy), homogeneity of space (momentum), and isotropy of space (angular momentum).
From this perspective it also makes sense why energy is "one-dimensional", since there is only one time dimension, and why momentum and angular momentum are "three-dimensional", since space has three dimensions.
However, why do the only additive integrals of motion correspond to these properties? What is special about them which guarantees that they have additive integrals of motion and that no other property can?
Even if you don't know the answer to all of these questions, I would really appreciate any help or insight you could give me. I was really enjoying this book until I thought of this question and now I am hopelessly confused. Thank you very much time and please enjoy the rest of your week!