Let us consider a classical mechanical system of N particles in a constant external field. We have 3N coordinates and 3N velocities, so totally 6N unknown variables. We have 6N ordinary differential equations of the first order for them or 3N equations of the second order. It is known that such a system has 6N-1 conserved quantities, see Landau-Lifshits, V. 1, Classical Mechanics, Chapter about Conservation Laws. In absence of symmetries these conserved quantities are rather complicated analytical expressions.

In case of some symmetries one can construct conserved quantities additive on particles. Everybody knows about total energy, momentum, and angular momentum conservation laws. But the number of conserved quantities remains to be 6N-1. It means the symmetries may help us construct "less messy" integrals of motion than in general case. We obtain simplification of analytical expressions but not new integrals of motion in case of symmetries.

The question I wrote in title has, in my opinion, the following answer: Symmetries do not lead to additional conservation laws but to simplification of existent ones. To a great extent it is due to simplification of equation system in case of symmetries.

An example of integrals of motion in 1D case:

1D Integrals of motion

  • $\begingroup$ But as I mentioned in a comment to you, integrals derived from Noether's theorem are not just "less messy". They are isolating integrals that define surfaces in phase space, provide useful informations about trajectories of the dynamical system and define integrable systems. They are qualitatively different. $\endgroup$ – Platypus Lover Feb 10 '11 at 22:59
  • 2
    $\begingroup$ What question are you asking? I don't see one other than an "I think ____, am I right?" question, which is discouraged in the FAQ. physics.stackexchange.com/faq $\endgroup$ – Mark Eichenlaub Feb 10 '11 at 23:20
  • 1
    $\begingroup$ @Vladimir: please stop making meaningless edits to your question, otherwise it'll be locked or deleted. $\endgroup$ – David Z Feb 11 '11 at 22:40
  • 2
    $\begingroup$ There seems to be a vicious circle here. People close the question early on, comments lead to edits in the question body which answer the question in effect, and then Vladimir has been accused that answers his question in the main body! , all because the question was closed ! The closing comment from below says:. "This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form" not that the question is answered at the same time it is asked! $\endgroup$ – anna v Feb 12 '11 at 8:33
  • 1
    $\begingroup$ I am also amazed at how often people are not aware of the very basic concept of "necessary" and "sufficient" . This is very common in climatology in which I have delved the past three years, but I would not expect it in a physics blog. $\endgroup$ – anna v Feb 12 '11 at 8:37