It has been widely known that the compressible Euler equation can be cast into the Hamiltonian form. For example, in the book "Dubrovin B A, Fomenko A T, Novikov S P. Modern geometry—methods and applications: Part II: The geometry and topology of manifolds[M]. Springer Science & Business Media, 2012", the Possion bracket is taken with variables $\rho, p_i$ (density and momentum) as $$\{p_i(x),p_j(y)\}=p_{j} (x){\partial \over \partial x_{i}} \delta(x-y) -p_i(y){\partial \over \partial x_{j}} \delta(x-y),$$ $$\{p_i(x),\rho(y)\} = \rho(x) {\partial \over \partial {x_i}} \delta(x-y).$$ And set $$H= \int \frac{p^2}{2\rho} + \epsilon(\rho)d^nx.$$ Then we have the Euler equations $${\partial \over \partial t} \rho =\{\rho,H\}= -\frac{\partial}{\partial x_j} p_j,$$ $${\partial \over \partial t}p_i = \{p_i,H\} = -\frac{\partial}{\partial x_j} (\frac{p_i p_j}{\rho})-\rho \partial_{x_j}\epsilon_\rho .$$ How to put this Hamiltonian structure into the symplectic form?

It seems that this non-canonical Hamiltonian structure makes it difficult to find the canonical coordinates $p,q$ as is often used in the symplectic formalism of Hamiltonian mechanics. In Arnold's book "Arnol'd V I. Mathematical methods of classical mechanics[M]. Springer Science & Business Media, 2013", there is an appendix illustrating the incompressible Euler equations as geodesics of left-invariant metric in Lie group. I was unable to understand the appendix, could anyone tells how the symplectic structure is involved here?

In general, the conservative system in physics can be seen as a Hamiltonian system. The fields system such as Hydrodynamics fields, the Einstein fields, the Yang-Mills fields are such examples. How to formulate the fields theory into symplectic form in general? Is there some books or classical papers answered this question?