# “Euler system” in Christodoulou's The Action Principle and PDEs

In The Action Principle and PDEs Christodoulou spends some time describing what he calls the Euler system associated to a system of variational PDEs (sections 2.5-7, 6.2). Briefly, given a bundle $E\to M$ with fiber $N$, whose sections are subjected to the variational PDE system, the Euler system is a system of equations whose solutions can be put into correspondence with special adapted coordinate systems $(x,q)$ on the total space $E$ of the bundle, where $x$ are coordinates on $M$ and $q$ are fiber coordinates whose level sets $q=q_0$ are all solutions of the variational PDE system. (Sanity check: codimension of the level set $q=q_0$ in $E$ is $\dim N$, which makes each such level set a section.) At least, that's my understanding.

The motivation for introducing the Euler system is appears to be that it is related to the so-called field theories of Carathéodory and De Donder-Weyl (later generalized by Le Page). The term "field theory" is a technical one in the theory of the calculus of variations and may not be familiar to everyone (I was certainly not familiar with it before looking into this). The goal of these, classical, field theories is to produce generalizations of Weierstrass's sufficient conditions for a strong extremum to variational problems with multiple independent variables.

Now my question:

What role does the Euler system play in Christodoulou's book?

The reason I ask is that I can't think of any. The main results of book, as I see them, are a formulation of the conditions for regular hyperbolicity of a system of variational PDEs and a proof of a domain of dependence theorem for regular hyperbolic systems. As far as I can see, every development in the book is eventually used to establish these main results, with the exception of the Euler system. In particular, I do not see Christodoulou make any attempt to use the solutions of the Euler system to establish any kind of strong extremum condition, in analogy with the works of Carathéodory, Weyl, etc.

So, am I missing something or is the Euler system really not used anywhere after being set up?

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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. – Willie Wong Nov 23 '11 at 11:23
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. – Igor Khavkine Nov 23 '11 at 12:12
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 $x$. – Willie Wong Nov 23 '11 at 13:54
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 – Willie Wong Nov 23 '11 at 14:23
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. – Willie Wong Nov 23 '11 at 16:02