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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?

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  • $\begingroup$ You might look at the book "Symplectic techniques in physics" by Guillemin and Sternberg for some examples. $\endgroup$ – Ben McKay May 23 '16 at 8:29
  • $\begingroup$ I am sorry that the book is not available at my university library. Could you recommend some other books or papers? Is there some more rigorous treatment from mathematicians? $\endgroup$ – sam May 23 '16 at 14:56
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In my opinion, you should attempt to understand the case of incompressible fluids before looking at the compressible case, since it's significantly more involved. You can find a discussion of the latter in "Semidirect Products and Reduction in Mechanics" by Marsden, Ratiu, and Weinstein.

For the incompressible case, the result is originally due to Arnold, so that's a good place to learn it. You could also try Section 1.5 of "Mechanics and Symmetry" by Marsden and Ratiu.

Edit: I should probably add a couple of comments to explain the main idea: the reason the Poisson bracket looks noncanonical is because it is induced by the Kostant-Kirillov-Souriau form on the coadjoint orbits of the volume-preserving diffeomorphism group (this is sometimes called the Lie-Poisson bracket). Although you can locally define canonical coordinates on these orbits, such coordinates aren't often geometrically meaningful (though it is possible to introduce canonically conjugate Clebsch variables for the problem - see here). The KKS symplectic form on coadjoint orbits of a group $G$ can be obtained by applying symplectic reduction to the $T^*G$ with its usual symplectic structure. Hence to see canonically conjugate variables, you need to look at the problem in $T^*Diff_\mathrm{vol}$, where $Diff_\mathrm{vol}$ is the group of volume-preserving diffeomorphisms. This is equivalent to considering the problem in Lagrangian coordinates, as opposed to the reduced picture, which considers the problem in Eulerian coordinates. This is all explained in the book "Mechanics and Symmetry" mentioned above.

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My first suggestion was:

MR1066693 (91d:58073) Reviewed
Guillemin, Victor(1-MIT); Sternberg, Shlomo(1-HRV)
Symplectic techniques in physics.
Second edition. Cambridge University Press, 
Cambridge, 1990. xii+468 pp. ISBN: 0-521-38990-9
58F05 (58-02 58F06 70Hxx)

It defines the symplectic structure of the Einstein equations and of charged particles in a Yang-Mills field.

I haven't read the following, but the review on Mathscinet suggests it defines the symplectic structures on the spaces of solutions of various partial differential equations in physics, including incompressible fluid flow, Korteweg-deVries and sine-Gordon:

MR0969603 (90f:58072) Reviewed
Schmid, Rudolf(1-EMRY)
Infinite-dimensional Hamiltonian systems.
Monographs and Textbooks in Physical Science. 
Lecture Notes, 3. Bibliopolis, Naples, 1987. 
viii+143 pp. ISBN: 88-7088-066-4
58F05 (35Q20 58D25)
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  • $\begingroup$ Thanks, though I am not accessible to these two books. There is a lecture note on symplectic geometry in my library by Sternberg, I think it may be helpful. $\endgroup$ – sam May 24 '16 at 2:49

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