A nonlinear Hamiltonian structure for the Euler equations
type: Item
instance of: scholarly article
title: A nonlinear Hamiltonian structure for the Euler equations
zbMATH Open document ID: 0534.76035
DOI: 10.1016/0022-247X(82)90100-7
author: Peter J. Olver
published in: Journal of Mathematical Analysis and Applications
publication date: 1982-01-01T00:00:00Z
review text: Using a characterization of the Hamiltonian operator previously given by the author [Math. Proc. Camb. Philos. Soc. 88, 71-88 (1980; Zbl 0445.58012)], it is shown that, in their natural Eulerian coordinates, the equations of inviscid incompressible fluid flow enjoy a Hamiltonian structure in the sense of the formal variational calculus. More precisely, for the vorticity equations (with u the velocity field, \(\omega =rot u\) the velocity): \(\partial \omega /\partial t=[u,\omega]\) such a structure is defined by the Hamiltonian operator \({\mathcal D}_{\omega}=-[\omega,rot\cdot]\) with Hamiltonian functional \(H=1/2\int | u|^ 2dx.\) In this framework, via the Noether theorem, the Lie symmetries independent of time of the Euler equations yield only the well known first integrals \(\int udx\), \(\int(x\times u)dx, \int | u|^ 2dx,\) while the total helicity \(\int<u,\omega>dx\) in 3 dimensions and the area integrals \(\int f(\omega)dx\) in dimension 2 seem to exhaust the conservation laws due to the degeneracy of the Hamiltonian operator. It is conjectured that there are no generalized Lie (or Lie-Bäcklund) symmetries for the Euler equations, hence that these would provide all the conservation laws whose densities are local functions in the Eulerian coordinates and their derivatives. This Hamiltonian structure is however known. Considered on the functional space of divergence free vector fields, the symplectic form \(\Omega_{\omega}(\xi,\eta)=<\xi,({\mathcal D}_{\omega})^{- 1}\!_{\eta}>\) (where \(<,>\) denotes \(L^ 2\) scalar product) is the direct image by the operator rot (that intertwines the coadjoint with the adjoint representation) of the Kirillov form \({\tilde \Omega}_ u(v,w)=<v,(\tilde {\mathcal D}_ u)^{-1}w>, \tilde {\mathcal D}_ uv=ad^*_ vu=-(ad_ v)^*u,\) well defined only on the coadjoint orbit of u [see \textit{V. Arnold}, Mathematical methods of classical mechanics (1978; Zbl 0386.70001), Appendix 2 ]. Thus the kernel of \({\mathcal D}_{rot u}\) (superfi…
Mathematics Subject Classification ID: 76B99
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 3848814
zbMATH Keywords: nonlinear Hamiltonian structure for Euler equations
zbMATH Keywords: coadjoint and adjoint representation
zbMATH Keywords: natural Eulerian coordinates
zbMATH Keywords: formal variational calculus
zbMATH Keywords: vorticity equations
zbMATH Keywords: Noether theorem
zbMATH Keywords: Lie symmetries independent of time
zbMATH Keywords: first integrals
zbMATH Keywords: total helicity
zbMATH Keywords: area integrals
zbMATH Keywords: conservation laws
zbMATH Keywords: symplectic form
zbMATH Keywords: Kirillov form
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3848814