Timeline for "Euler system" in Christodoulou's The Action Principle and PDEs
Current License: CC BY-SA 4.0
10 events
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| S Jul 17, 2022 at 10:34 | history | suggested | The Amplitwist | CC BY-SA 4.0 |
fixed broken link to eom.springer.de
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| Jul 17, 2022 at 9:53 | review | Suggested edits | |||
| S Jul 17, 2022 at 10:34 | |||||
| Nov 23, 2011 at 16:02 | comment | added | Willie Wong | Going back to the final sentence in your question: I believe the answer is "no, you didn't miss any thing." In which case, (empirically speaking) I think you are probably not the only one who have "yet to figure out how a solution of the Euler system would be useful in either a PDE or physics context" :p. | |
| Nov 23, 2011 at 14:57 | comment | added | Igor Khavkine | Right. I believe that the property of the Euler system that you point out is summarized in the first paragraph of my question. Though I've still yet to figure out how a solution of the Euler system would be useful in either a PDE or physics context, hence my question. Especially, since, as you've also pointed out, the analogy with the work of Weyl, Carathéodory and Weierstrass for elliptic variational problems is not applicable to the hyperbolic case. | |
| Nov 23, 2011 at 14:28 | comment | added | Willie Wong | point a well-posed problem. In the context of exterior differential systems, there are some known results in this general direction (Deane Yang would know much more). (Theorem 2.8, in the case of one independent variable, is not that interesting, since it is just the fundamental theorem of ODEs. But try thinking about it in cases with more than one independent variables.) In any case, my own understanding of this aspect of the book is still very murky right now. So I hope what I said didn't just add to the confusion. | |
| Nov 23, 2011 at 14:23 | comment | added | Willie Wong | Velocity space is $(t,x,v)$, and phase space is $(t,x,p)$. The Hamiltonian system studies $t\to (t,x,p)$, while the Euler system looks at $(t,x) \to (t,x,p)$. Christodoulou explains this as the fluid system related to the Newtonian system. What's interesting, I think, is Theorem 2.8 in the book. Imagine trying to solve a PDE by taking the value of the solution at one single point and integrating. This generally cannot be done because of lack of uniqueness. Theorem 2.8 is like saying that solving the Euler system gives you enough information to make the problem of posing initial data at one | |
| Nov 23, 2011 at 13:54 | comment | added | Willie Wong | Hmm... it appears on re-reading, what I wrote above does not accurately reflect what I wanted to say. On a very formal level, the mapping from Lagrangian to Hamiltonian is what we are familiar with: mapping velocity $v$ to conjugate momentum $p$. In both pictures solutions are considered as sections over the independent variables (so in Newtonian mechanics, the independent variable is just time). The Euler systems picture, however, studies sections of the phase space over configuration space. In the Newtonian motion of one particle, the independent variable is $t$, dependent is $x$. | |
| Nov 23, 2011 at 12:12 | comment | added | Igor Khavkine | Wille, thanks for your comment! I think your last statement is getting at the point I am trying to understand. Perhaps you could elaborate on it. Which flows do you mean? And what is the connection do you see developed between the Lagrangian and Hamiltonian pictures? I have my own understanding of some of these things, from which the development of the Euler system appears to be superfluous. But maybe I just have trouble seeing it from a different point of view. | |
| Nov 23, 2011 at 11:23 | comment | added | Willie Wong | One note, since the focus of the book is on hyperbolic PDEs, you shouldn't expect anything like Weierstrass's conditions. Solutions of hyperbolic PDEs from action principles tend to be non-extremal critical points (just think about the trivial solution to the linear wave equation). Also, I feel that the mention of the Euler system is an end in itself: it develops the connection between Lagrangian and Hamiltonian points of view, especially showing how one can derive the flows associated to the system. | |
| Nov 22, 2011 at 11:40 | history | asked | Igor Khavkine | CC BY-SA 3.0 |