In [The Action Principle and PDEs][1] 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][2] 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?

  [1]: https://books.google.com/books?id=az7xm9vuhwIC
  [2]: https://encyclopediaofmath.org/wiki/Weierstrass_conditions_(for_a_variational_extremum)