A nonlinear Hamiltonian structure for the Euler equations
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 \({\mathcal D}_{rot u}\) (superficially examined in the paper) corresponds by rot with the kernel of \(\tilde {\mathcal D}_ u\), hence with the orthocomplement of the tangent space to the coadjoint orbit.
nonlinear Hamiltonian structure for Euler equations
scientific article; zbMATH DE number 3848814
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