Relationship between symmetries and additive integrals of motion

Question

I'm currently reading Landau and Lifshitz's Statistical Physics. In it, they attempt to justify that the density function only depends on the energy by arguing that the logarithm of this function is additive, and there are, at most, "only 7 seven independent additive integrals of motion". While looking for the answer to this claim, I found this post and this one.

It appears that their claim is not quite correct, and that in fact there are 10 such integrals of motion. The first post I linked states that these 10 integrals of motion correspond to the 10 generators of the Galileo group, and I'd like to know more about this correspondance.

First of all, I suppose that the independence between the constants of motion comes from the fact that they are associated with the generators of the group. Is that supposition correct? If so, how exactly does that come about?

Secondly, is it just a coincidence that these integrals are additive? Or is it that some property of the Galileo group ensures that their associated constants of motion are additive?

Thirdly, would swapping the Galileo group by the Poincaré group change anything in these considerations? From what I could gather, it still has 10 generators, but are their associated integrals of motion additive as well?

Note that I'm using additive to mean that if the system under consideration is broken into $n$ subsystems, each with the integral of motion associated with the same symmetry being $f_i$, then the integral of motion of the entire system, associated with that same symmetry, will be $$f=\sum_{i=1}^nf_i$$

Answer 1

The following is a brief overview of the key points you ask about. Many of the specifics have already been discussed in the linked posts.  

1. Independence of Constants of Motion: In the context of Lie groups, each generator $X_i$ corresponds to a distinct symmetry transformation. The independence of these generators implies that no $X_i$ can be expressed as a linear combination of other generators. In physics, this means each conserved quantity $C_i$ (derived via Noether's theorem) is independent, as they correspond to different symmetries.

2. Additivity of Integrals of Motion: The additivity of integrals of motion, represented as $f = \sum_{i=1}^n f_i$, arises from the linear nature of the symmetry group's generators. These generators correspond to linear transformations, making the associated conserved quantities naturally additive over subsystems.

3. Galileo vs. Poincaré Group: Both groups have 10 generators, but their interpretations differ (Galilean boosts in Galileo group vs. Lorentz boosts in Poincaré group). The integrals of motion remain additive in both cases, reflecting the fundamental symmetries of their respective spacetimes. However, the specific forms of these integrals differ, influenced by the relativistic aspects of the Poincaré group compared to the non-relativistic Galileo group.