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Poisson brackets in hydrodynamics. (English) Zbl 1139.53040
This paper discusses various Poisson structures which provide the Hamiltonian formulations of numerous evolution equations in fluid mechanics. The principal observation is that in infinite dimensions the Lie-Poisson brackets are not defined for all smooth functionals. This rises the following questions: whether the chosen bracket is closed for the class of the functionals for which it is defined and whether the fundamental Jacobi identity is satisfied? The presented discussion based on concrete examples and the supplied list of references will be of definite help for the readers who are interested in peculiarities of Lie-Poisson structures in infinite dimensional spaces.
MSC:
53D17Poisson manifolds; Poisson groupoids and algebroids
37K05Hamiltonian structures, symmetries, variational principles, conservation laws
37K65Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics