Timeline for Practical example of Hamiltonian reduction
Current License: CC BY-SA 4.0
17 events
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| 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 9:04 | comment | added | Konstantinos Kanakoglou | Yes, in this case, both methods are in the end the same thing. The Lie-Cartan method is a meaningful generalization, if we have more than one, non-commutative integrals. | |
| Jun 6, 2019 at 7:33 | comment | added | Doriano Brogioli | Thank you. My question is only about the very simple case of $H$ with a single constant of motion $J$. If I'm not wrong, the Lie-Cartan theorem has nothing to say in this case. Can you confirm this? | |
| Jun 6, 2019 at 1:26 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| Jun 6, 2019 at 1:05 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| Jun 6, 2019 at 0:35 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| Jun 6, 2019 at 0:21 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| Jun 6, 2019 at 0:00 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| Jun 5, 2019 at 23:58 | comment | added | Konstantinos Kanakoglou | i have edited, hoping to make this point more clear. | |
| Jun 5, 2019 at 23:54 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| Jun 5, 2019 at 8:38 | comment | added | Doriano Brogioli | Yes, it is an example. This answers my question. Why do you say that the Lie-Cartan method is less restrictive when applied globally? | |
| Jun 5, 2019 at 0:36 | comment | added | Konstantinos Kanakoglou | what about example 3.13? doesn't it provide an explicit case of what you are asking for (in the sense that the reduction applies globally) ? | |
| Jun 4, 2019 at 21:20 | comment | added | Konstantinos Kanakoglou | Yes you are right, the description as stated in the proposition is local. But generally, it depends on the domain of the integrals; i think this is essentially the same thing, since the Liouville integrability, also applies in an open set containing the level set (i.e. the set at which the integrals have fixed values). if the level set is the whole of the phase space then it applies globally. I think that the Poincare reduction and the Lie-Cartan method also do; furthermore the Lie-Cartan method is less restrictive than Poincare reduction (when applied globally). | |
| Jun 4, 2019 at 20:16 | comment | added | Doriano Brogioli | If I correctly understand, proposition 3.2 is only local. The theorem about Liouville integrability instead is global (as long as the integrals are defined globally). Actually, this makes sense, since it is easy to find an $F$ such that there is no transformation under which $F=y_1$ globally. Do you confirm this, that the property is only valid locally? | |
| Jun 4, 2019 at 19:41 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| Jun 4, 2019 at 16:44 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| Jun 4, 2019 at 16:26 | history | answered | Konstantinos Kanakoglou | CC BY-SA 4.0 |