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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

Vladimir Kalitvianski
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  • 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. – Platypus Lover Feb 10 '11 at 22:59
  • I completely agree that they are useful. The question is not here. Some people think that symmetries lead to conservation laws, that without symmetries there is no conservation of anything. – Vladimir Kalitvianski Feb 10 '11 at 23:03
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    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. http://physics.stackexchange.com/faq – Mark Eichenlaub Feb 10 '11 at 23:20
  • Mark, the question is in the title. Yes, I answer it according to my knowledge, but if you look aside, they downvote my answers severely and without explanations. So I would like to see their arguments. I would like to learn. – Vladimir Kalitvianski Feb 10 '11 at 23:25
  • Hm, but when you break symmetries, then conservation is often invalidated as well. Example: A crystal lattice. Here, translation invariance is broken and this breaks conservation of momentum: Scattered light has its momentum conserved only modulo the reciprocal lattice vectors – Lagerbaer Feb 10 '11 at 23:29
  • When the symmetry is broken, the conservation laws become more complicated. Symmetry simplifies the motion laws but any motion is "integrable" in the sense of existence of solutions. – Vladimir Kalitvianski Feb 10 '11 at 23:33
  • Now that I'm thinking about this, isn't the Noether Theorem reversible, i.e., for every symmetry there's a conserved quantity and vice versa? In this case, symmetries and conserved quantities are somewhat the same. – Lagerbaer Feb 11 '11 at 00:18
  • @Mark: as far as I can tell, it doesn't say anything in the FAQ about those kinds of questions being disallowed in general. (Although in practice, they do have a tendency to be less well posed than an average question, but that has to be judged on a case-by-case basis.) – David Z Feb 11 '11 at 00:22
  • @Vladimir: It would be a big help if you edit the question body to include your actual question. The title is really supposed to be just that, a title, a "teaser" of sorts to draw people in. A well written question can be understood without even looking at the title. – David Z Feb 11 '11 at 00:25
  • @Lagerbaer, Noether's theorem is definitely not reversible! – wsc Feb 11 '11 at 00:55
  • To David: Thanks, David, for your suggestion. I rarely get a valuable and kind advice. – Vladimir Kalitvianski Feb 11 '11 at 08:36
  • To Lagerbaer: no, there are conservations laws without any symmetry (intergals of motion) and there are conserved quantities that are conserved by definition such as masses and charges (phenomenological parameters). The latter are not functions of dynamical variables and have nothing to do with the Noether theorem at all, to tell the truth. – Vladimir Kalitvianski Feb 11 '11 at 10:36
  • @Vladimir, you have an unambiguous question in the tite, then you answer it! Effectively, you're getting people to comment via their answers to your answer in the question, which is getting your questions closed. – John McAndrew Feb 11 '11 at 16:31
  • To John: this question was closed by some friends of science just because they "did not find a question in it". – Vladimir Kalitvianski Feb 11 '11 at 17:27
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    @Vladimir: please stop making meaningless edits to your question, otherwise it'll be locked or deleted. – David Z Feb 11 '11 at 22:40
  • David, I simplified formulas and gave explicit solutions. – Vladimir Kalitvianski Feb 11 '11 at 22:44
  • @Vladimir: ah, it doesn't show those changes in the normal diff view. Still, you've made a lot of edits to this question, and there have been complaints about it continually resurfacing on the front page of the site. Please limit yourself to major edits from now on. (Also, if and when you do make a major edit, please remove the image and include that content as text) – David Z Feb 11 '11 at 23:23
  • @Vladimir, I'm on your side, but it's true that there doesn't seem to be any question in the text of what you've written. Why not post another titled "Do symmetries simplify existing conservations laws in general?". Give some examples where this is true and then end it with the question if it's generally true. That way you give your points, and conclude it with a question that you haven't answered. – John McAndrew Feb 11 '11 at 23:36
  • Dear John, I asked and answered another, more important question. The symmetries simplify things by definition. It's not really interesting to demonstrate. – Vladimir Kalitvianski Feb 11 '11 at 23:51
  • @Vladimir, well be prepared to have more of your questions closed if you ask a question, only to answer it yourself. – John McAndrew Feb 12 '11 at 01:19
  • Jogn, I not only answer "myself" but also let the others answer. It is not me who close my questions and prevent the others from answering! – Vladimir Kalitvianski Feb 12 '11 at 08:20
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    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! – anna v Feb 12 '11 at 08:33
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    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. – anna v Feb 12 '11 at 08:37
  • I did warn you about making minor edits to this question. – David Z Apr 18 '11 at 18:02

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