Timeline for Translation of Marsden-Weinstein-Meyer into classical mechanics language
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| May 28, 2020 at 9:28 | history | edited | Ben McKay | CC BY-SA 4.0 |
added harmonic oscillator
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| May 28, 2020 at 8:59 | comment | added | Doriano Brogioli | Fine. So there are cases in which J cannot be brought into p_n by a global canonical transformation, but still MWM gives a global reduction. This reduction is expressed in terms of symplectic structure (I guess, by giving a suitable bilinear form) but not in terms of canonical transforms. I'm still kindly challenging you to give an example of such a J. | |
| May 28, 2020 at 8:43 | comment | added | Ben McKay | I have tried to add some detail to clarify the local versus global aspect. The conclusion is not just local; the manifold $Y$ of flow lines has a global symplectic structure, and the map $\varphi$ matches up the symplectic flow of $H$. | |
| May 28, 2020 at 8:41 | history | edited | Ben McKay | CC BY-SA 4.0 |
more detail on local versus global aspects
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| May 28, 2020 at 8:28 | comment | added | Doriano Brogioli | So MWM is only locally valid? Well, this is surprising! For this reason. In Arnold's books, it is proven that Poincarè's reduction can be done locally for any conserved quantity J. Here, instead, you are requiring in the hypothesis that J has a property: "the flow lines on X_J0 can be parametrized by a smooth manifold Y of dimension one less than the dimension of the level set." This is not true for any J (I have an example). So it seems that MWM hypothesis is more restrictive, while the thesis is the same. Is this true? | |
| May 28, 2020 at 6:49 | history | edited | Ben McKay | CC BY-SA 4.0 |
added Poincare reduction
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| May 27, 2020 at 20:12 | comment | added | Doriano Brogioli | Wonderful. Can you please add this statement in your answer? I would be grateful if you can also provide a reference where this is stated explicitly. I never found it. Btw, most mathematicians are not able to give this simple answer, so, yes, the question and the answer are not too trivial for MO. | |
| May 27, 2020 at 20:04 | comment | added | Ben McKay | In cases when your Lie group is 1-dimensional and simple connected, i.e. the real number line, i.e. when there is precisely one function $J$ as the moment map, i.e. the cases you want to know about, then MWM is Poincare reduction. | |
| May 27, 2020 at 20:03 | history | edited | Ben McKay | CC BY-SA 4.0 |
added simple example
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| May 27, 2020 at 20:03 | comment | added | Doriano Brogioli | Phi is not a canonical transformation, but the whole procedure can be described in the following way. You make a canonical transformation which brings J into p_n. Then, you consider H as a function of p_1, p_n-1 , q_1, .. q_n-1 ; you treat p_n as a fixed parameter and neglect q_n. This is the procedure called "Poincarè reduction" in Arnold's books. Here, I'm (kindly) challenging you to provide an example of H and J that can be treated with MWM but not with the Poincarè reduction. | |
| May 27, 2020 at 20:01 | history | edited | Ben McKay | CC BY-SA 4.0 |
added simple example
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| May 27, 2020 at 19:56 | comment | added | Ben McKay | It is not a canonical transformation, because the number of variables in $h$ is smaller than in $H$. Canonical transformations preserve dimension. But as I said, for the problem as you posed it, without Lie groups and with only one real valued function $J$, the result of MWM is just the obvious result above, when written in the appropriate Darboux coordinates. | |
| May 27, 2020 at 19:53 | comment | added | Doriano Brogioli | Naively, I would describe this procedure saying that we make a canonical transformation such that one of the Darboux coordinates is equal to J; this is the case in your example, or e.g. when we exploit the conservation of angular momentum in three-body problem. Instead you say that it it is not so easy and it requires more complex mathematical structures. Are you sure that there is at least one case of H and J in which the MWM holds, but the result cannot be obtained with a properly chosen canonical transformation? | |
| May 27, 2020 at 19:27 | history | edited | Ben McKay | CC BY-SA 4.0 |
added 108 characters in body
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| May 27, 2020 at 19:25 | comment | added | Ben McKay | @DorianoBrogioli: correct, I will fix that. | |
| May 27, 2020 at 19:19 | comment | added | Doriano Brogioli | There is still something unclear. It seems that Y parametrizes the flow lines only on a given level set (in the example in Darboux coordinates, the level set p_n=P has dimension n-1, and the flow lines on it are parametrized with n-2 dimensions). Then, phi should be a function from the level set to Y, not X -> Y. Could you please explain? | |
| May 27, 2020 at 18:58 | history | edited | Ben McKay | CC BY-SA 4.0 |
explained why Darboux coordinates make everything obvious
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| May 27, 2020 at 17:13 | history | edited | Ben McKay | CC BY-SA 4.0 |
added definition of regular value and of descent of a function and comment that this is not a suitable question
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| May 27, 2020 at 17:10 | comment | added | Michael Engelhardt | Oh - seems like what it's saying is that one can ignore an ignorable coordinate (using its conserved conjugate momentum as an input parameter) ... but maybe I'm being too simple-minded ... | |
| May 27, 2020 at 16:45 | comment | added | Doriano Brogioli | Could you please extend a bit? "Regular value" is a value that is not - say - a maximum or minimum? "Descends" means that you take the values of H on the manifold? "For a natural symplectic structure" means that there is a canonical transformation such that this happen? Yes, we physicists are simple minded people! | |
| May 27, 2020 at 16:21 | history | answered | Ben McKay | CC BY-SA 4.0 |