# Integrals of motion some more important than others?

1. Oct 18, 2011

### Master J

I've been reading Mechanics of Landau Lifgarbagez. They state that "not all integrals of motion are of equal importance", and that "there are some whose constancy is of profound importance"...these ones are conserved for the motion.

What confuses me is that I thought that's what an integral of the motion was!! A function of the phase space coordinates that was conserved for the motion! What i seeem to be reading is that integrals of motion are conserved, but some are more important because they are conserved???? Can someone clarify this?

On a similar point, for a single free particle with (2s - 1) = 5 integrals of motion, with s being the degrees of freedom, what are its integrals of motion? I would reason them as:
1) Linear momentum in x direction.
2) As above for y.
3) As above for z.
4) Total energy.
5) Angular momentum.

If that is the case, the howcome the angular momentum only takes one integral of motion? Surely the angular momentum vector can be expressed in terms of its x,y and z components?

Edit: I think i just solved the last part for myself....only the TOTAL angular momentum is conserved..unlike linear, where its individual components are each conserved...is that correct?
Any input helping to clear this up for me is highly appreciated!!
Thanks!

2. Oct 18, 2011

### QuArK21343

I think Landau here is not saying that some integrals of motion are more important because they remain constant throughout the motion (this is valid, by definition, for any integral of motion), but that there are some such that the fact itself that they are constants is of special theoretical importance. For example, some integral of motion may be due to a particular symmetry in the lagrangian of the system (any symmetry gives rise to an integral of motion). Among them, linear momentum is linked to space homogeneity, angular momentum to space isotropy and energy to time invariance. There may be many other integrals of motion, but they may not have such an important physical interpretation. Hope it helps