Timeline for Translation of Marsden-Weinstein-Meyer into classical mechanics language
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| May 29, 2020 at 13:32 | comment | added | Aaron Bergman | It sounds like you want to ask the question: is there an intrinsic characterization of those conserved quantities that give rise to a complete Hamiltonian vector field. Maybe ask a new question to make that clear? | |
| May 29, 2020 at 8:57 | comment | added | Doriano Brogioli | Yes, it sounds simple also to me. But I'm asking what it implies in terms of classical mechanics. The class of the J such that their Hamiltonian flow has the required property of "acting nicely" is clearly characterized. But now I'm asking: is it a class already known in classical mechanics, e.g. is it the class of integrable or superintegrable Hamiltonians, or can it be characterized by looking at the Lyapunov exponents, or has chaotic trajectories? Or it is completely unrelated? It is curious that the mathematicians find this question so strange and that there is no literature! | |
| May 28, 2020 at 21:03 | comment | added | Aaron Bergman | The theorem is stated in terms of group actions, ie, your conserved quantity is related to a continuous symmetry. That seems pretty simply to me. Probably the condition is just that the Hamiltonian vector field associate with J is complete, but I might be forgetting something. | |
| May 28, 2020 at 19:16 | comment | added | Doriano Brogioli | "the symmetry acts nicely on that manifold so the quotient is also a manifold" ... For example, it "acts nicely" for J=p_n, but it "does not act nicely" if J is a non-integrable Hamiltonian. Is this correct? Can you try to express the condition in terms of simpler things like integrability? | |
| May 28, 2020 at 17:58 | comment | added | Aaron Bergman | The condition is that the value of the conserved variable is a regular value, which means that the inverse image is a manifold. You also have to assume that the symmetry (sticking to the single conserved quantity case) acts nicely on that manifold so the quotient is also a manifold. | |
| May 28, 2020 at 14:57 | comment | added | Doriano Brogioli | The hypothesis you mention is on the Hamiltonian flow of J? @Ben McKay says that "the set of flow lines are parameterized by a smooth manifold Y of dimension one less than the dimension of the level set." Is this the hypothesis? Is so, please add it to your answer. | |
| May 28, 2020 at 14:16 | comment | added | Aaron Bergman | Yes, it's global. The hypotheses are mostly to ensure that the inverse image is a nice space that is acted on nicely by the residual symmetries. | |
| May 28, 2020 at 7:53 | comment | added | Doriano Brogioli | This is called "Poincarè reduction" in the Arnold's books. It is demonstrated that it can be done locally. I was wondering if MWM can say something more on the topic. So, is MWM globally valid? And what are exactly the hypothesis on the conserved value? | |
| May 28, 2020 at 2:56 | history | answered | Aaron Bergman | CC BY-SA 4.0 |