Timeline for Practical example of Hamiltonian reduction
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17 events
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| Jun 8, 2019 at 10:58 | comment | added | Doriano Brogioli | If I correctly understand, in that case, the reduction is more complex than in the usual case of Poincare'. So we cannot perform the usual Poincare' reduction globally, but the other reduction is feasible globally. Is there any theorem on this? I mean: a theorem which guarantees the existence of a global reduction (in which the reduction takes a more general form than the Poincare' one)? | |
| Jun 7, 2019 at 20:42 | comment | added | Francois Ziegler | @DorianoBrogioli Note that my concluding item already gives a counterexample of the sort you are after, if we include nonnegative energies: the reduced space changes topology, so I think no “global coordinates” will cover both regions. | |
| Jun 7, 2019 at 20:32 | history | edited | Francois Ziegler | CC BY-SA 4.0 |
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| Jun 7, 2019 at 20:27 | history | edited | Francois Ziegler | CC BY-SA 4.0 |
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| Jun 6, 2019 at 21:29 | comment | added | Doriano Brogioli | Let us continue this discussion in chat. | |
| Jun 6, 2019 at 20:52 | vote | accept | Doriano Brogioli | ||
| Jun 6, 2019 at 15:45 | comment | added | Doriano Brogioli | After getting the answers, I clarified myself the problem and I opened a new question: mathoverflow.net/questions/333409/… . If you are interested, please have a look. Thank you again for the help. | |
| Jun 6, 2019 at 14:24 | comment | added | Francois Ziegler | I don’t know... Anyway, traditionally reduction (your title) means passage to the leaf space of a level set, not to other (global? what for?) coordinates on the original manifold; and the point of Marsden-Weinstein is to provide a priori conditions under which that works. (Compare the number of times Arnol’d must say “if” as he uses the Lie-Cartan Theorem, 3.16, to “reduce the order”.) | |
| Jun 6, 2019 at 11:25 | comment | added | Doriano Brogioli | Very good, thank you. So Liouville's integration is ensured to hold globally, provided a non-degeneracy condition. Do you know if there is a similar condition (or any condition) making the Poincare' method globally valid? | |
| Jun 6, 2019 at 10:46 | comment | added | Francois Ziegler | Arnol’d assumes that the hamiltonian vector fields (1) of his commuting functions are everywhere linearly independent, in particular nonzero. When this fails he only guarantees action-angle variables in a level’s neighborhood (Section 50, Remark 3). Back to one first integral, indeed his (2006, §3.2.2) gives (again) local coordinates on one of which $H$ does not depend. (Marsden-Weinstein is in that section, 3 pages later.) | |
| Jun 6, 2019 at 9:44 | comment | added | Doriano Brogioli | For example, Arnold, "Mathematical methods of classical mechanics", chapter 10 section 49, the proof of the validity of Liouville's integration is global, while in Arnold et al., "Mathematical Aspects of Classical and Celestial Mechanics", proposition 3.2, the proof of the validity of Poincare' method is local. | |
| Jun 6, 2019 at 9:18 | comment | added | Doriano Brogioli | The citation of Duistermaat makes me confused. The proofs of Liouville's integrability that I saw do not rely on the selection of a small neighborhood, thus I assumed they are global. Instead, the proof of the validity of the Poincare' method is critically local. In other words, I can easily find a constant of motion $\Psi$ which cannot be used for the Poincare' reduction, but I never saw a set of $n$ constants of motion in involution not giving rise to the Liouville integration. | |
| Jun 6, 2019 at 8:27 | comment | added | Francois Ziegler | @DorianoBrogioli Anything to do with coordinates (as your question asks for) is ipso facto only local. So you can take $\psi$ as a local coordinate — but only away form its critical points (where (1) vanishes; the “free action” condition excludes such points). Likewise the Darboux charts I mention are only local canonical transformations. Whether this can all be made global is a much subtler question, see e.g. Duistermaat (1980). | |
| Jun 6, 2019 at 7:42 | comment | added | Doriano Brogioli | Thank you for contributin. However, I would like to ask you a "translation" in a simpler language. I understand that the presence of $\Psi$ decreases the space dimension to $2n-2$. Is this valid locally or globally? You say "taking $\Psi$ as $q_n$": do you really mean that the constant of motion $\Psi$ becomes equal to one of the coordinates, $q_n$, upon a canonical transformation? | |
| Jun 6, 2019 at 7:00 | history | edited | Francois Ziegler | CC BY-SA 4.0 |
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| Jun 6, 2019 at 6:31 | history | edited | Francois Ziegler | CC BY-SA 4.0 |
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| Jun 6, 2019 at 6:26 | history | answered | Francois Ziegler | CC BY-SA 4.0 |