Timeline for Understanding Hamilton's equations in classical field theory in a rigorous way
Current License: CC BY-SA 4.0
23 events
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| S Feb 12, 2021 at 21:06 | history | bounty ended | CommunityBot | ||
| S Feb 12, 2021 at 21:06 | history | notice removed | CommunityBot | ||
| Feb 6, 2021 at 17:30 | answer | added | Arnold Neumaier | timeline score: 2 | |
| S Feb 4, 2021 at 19:28 | history | bounty started | MathMath | ||
| S Feb 4, 2021 at 19:28 | history | notice added | MathMath | Draw attention | |
| Feb 3, 2021 at 12:19 | comment | added | Cosmas Zachos | On this SE, of course... | |
| Feb 2, 2021 at 23:09 | comment | added | MathMath | Precisely. But my point is: I can simply define the Hamiltonian in infinite-dimensional space by (6), mimicking the finite-dimensional case or I could define Legendre transforms in infinite-dimensional spaces and show that the Hamiltonian, defined as this Legendre transform, becomes (6). Is the second approach doable? | |
| Feb 2, 2021 at 21:53 | comment | added | Cosmas Zachos | One generalizes the finite variables' expression to one with an infinity of variables, mutatis mutandis. | |
| Feb 2, 2021 at 21:50 | comment | added | MathMath | I think this point is. But the first question is still bothering me. | |
| Feb 2, 2021 at 21:45 | comment | added | Cosmas Zachos | OK, is all clear now? | |
| Feb 2, 2021 at 21:43 | comment | added | MathMath | @CosmasZachos I understand it as a functional derivative, in fact. But there seems to be some sort of misunderstandings between the treatment of functional derivatives as it is used by physicists and mathematicians. As it is the case in the linked post, physicists use functional derivatives as directional derivatives in the direction of delta distributions, while mathematicians define it as Gatêaux derivatives. It seems that what physicists do is to find the integral kernel of the functional, while mathematicians use auxiliary theorems to prove this kernel satisfies the wanted equations. | |
| Feb 2, 2021 at 21:09 | comment | added | Cosmas Zachos | Linked. | |
| Feb 2, 2021 at 21:03 | comment | added | Cosmas Zachos | "What is the meaning of the derivative in the right hand side of (4)?" You do understand it as a functional derivative, so pretty much (2), upon the possibly confusing notational switch ${\mathbf x}\mapsto \phi ({\mathbf x}), ~~{\mathbf p}\mapsto \pi ({\mathbf x})$, no? | |
| Feb 2, 2021 at 12:00 | history | tweeted | twitter.com/StackPhysics/status/1356573105393655814 | ||
| Feb 2, 2021 at 2:59 | history | edited | MathMath | CC BY-SA 4.0 |
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| Feb 2, 2021 at 1:20 | comment | added | MathMath | Also, in fact Marsden et. al. have a beautiful set of books on the topic. Much of the material, however, is discussed using manifolds and tangent bundles, which I'm not familiar yet. So I'm looking for a more 'functional analysis' approach, if I may say so. | |
| Feb 2, 2021 at 1:18 | comment | added | MathMath | @Qmechanic thanks for the comments. About eq. (2), you mean that this is not one of Hamilton's equations because it is usually stated as the definition of ${\bf{p}}$, right? This is what I meant, but it sounds wrong in my post. You are right. About the post linked: this is exactly what I'm looking for but for a field theory instead of point mechanics! | |
| Feb 2, 2021 at 1:16 | comment | added | MathMath | @DanielC yes, Arnold do not discuss it sadly. I followed Arnold only in the section about Legendre transforms for finitely many variables. | |
| Feb 2, 2021 at 1:14 | comment | added | DanielC | Off the top of my head, I do not recall Arnold discussing classical field theory, but Abraham, Marsden, and Ratiu do. And Giaccheta and Sardanshvili wrote two monographs on mathematical field theory. One of these three references should a rigorous definition of a Legendre transformation of the "configurations space". | |
| Feb 2, 2021 at 1:05 | comment | added | Qmechanic♦ | Related question in point mechanics: physics.stackexchange.com/q/105912/2451 | |
| Feb 2, 2021 at 1:04 | history | edited | Qmechanic♦ |
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| Feb 2, 2021 at 0:55 | comment | added | Qmechanic♦ | Comment to the post (v2): Eq. (2) is not one of the Hamilton's equations per se. | |
| Feb 2, 2021 at 0:40 | history | asked | MathMath | CC BY-SA 4.0 |