A simple way of making a Hamiltonian system into a bi-Hamiltonian one

Abstract

Given a Poisson structure (or, equivalently, a Hamiltonian operator) $P$, we show that its Lie derivative $L_{\tau }(P)$ along a vector field $\tau$ defines another Poisson structure, which is automatically compatible with $P$, if and only if $[L_{\tau }^2(P),P]=0$, where $[\cdot ,\cdot ]$ is the Schouten bracket. We further prove that if $\dim \ker P\le 1$ and $P$ is of locally constant rank, then all Poisson structures compatible with a given Poisson structure $P$ on a finite-dimensional manifold $M$ are locally of the form $L_{\tau }(P)$, where $\tau$ is a local vector field such that $L_{\tau }^2(P)=L_{\tilde{\tau }}(P)$ for some other local vector field $\tilde{\tau }$. This leads to a remarkably simple construction of bi-Hamiltonian dynamical systems. We also present a generalization of these results to the infinite-dimensional case. In particular, we provide a new description for pencils of compatible local Hamiltonian operators of Dubrovin--Novikov type and associated bi-Hamiltonian systems of hydrodynamic type. Key words: compatible Poisson structures, Hamiltonian operators, bi-Hamiltonian systems (= bihamiltonian systems), integrability, Schouten bracket, master symmetry, Lichnerowicz--Poisson cohomology, hydrodynamic type systems. MSC 2000: Primary: 37K10; Secondary: 37K05, 37J35

arXiv comment12 pages, LaTeX 2e, no figures, accepted for publication in Acta Appl. Math. Major revision: In this version an important condition of local constancy of rank of P is added (it is assumed that the vicinities where rank P=const are of the same dimension as the underlying manifold M). Moreover, this version contains Remarks 1 and 2, references [14],[22],[23],[29],[30],[36],[41], and the discussion thereof that for technical reasons were not included in the published version of the paper

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