A simple way of making a Hamiltonian system into a bi-Hamiltonian one
Abstract
Description
Given a Poisson structure (or, equivalently, a Hamiltonian operator) $P$, we show that its Lie derivative $L_τ(P)$ along a vector field $τ$ defines another Poisson structure, which is automatically compatible with $P$, if and only if $[L_τ^2(P),P]=0$, where $[\cdot,\cdot]$ is the Schouten bracket. We further prove that if $\dim\ker P\leq 1$ 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 $L_τ(P)$, where $τ$ is a local vector field such that $L_τ^2(P)=L_{\tildeτ}(P)$ for some other local vector field $\tildeτ$. 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
12 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
12 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