This article is cited in 2 papers

Dispersive deformations of the Hamiltonian structure of Euler's equations

M. Casati

Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy

Abstract: Euler's equations for a two-dimensional fluid can be written in the Hamiltonian form, where the Poisson bracket is the Lie–Poisson bracket associated with the Lie algebra of divergence-free vector fields. For the two-dimensional hydrodynamics of ideal fluids, we propose a derivation of the Poisson brackets using a reduction from the bracket associated with the full algebra of vector fields. Taking the results of some recent studies of the deformations of Lie–Poisson brackets of vector fields into account, we investigate the dispersive deformations of the Poisson brackets of Euler's equation: we show that they are trivial up to the second order.

Keywords: Euler's equations, Poisson bracket, Poisson vertex algebra.

MSC: 37K05 (primary), 76M60, 17B80

DOI: 10.4213/tmf9042

 English version: Theoretical and Mathematical Physics, 2016, 188:3, 1296–1304

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