A simple way of making a Hamiltonian system into a bi-Hamiltonian one
| dc.creator | Sergyeyev, A. | |
| dc.date | 2003-10-13 | |
| dc.date | 2004-04-19 | |
| dc.date.accessioned | 2026-07-07T05:35:00Z | |
| dc.date.available | 2026-07-07T05:35:00Z | |
| dc.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 | |
| dc.description | 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 | |
| dc.identifier | https://arxiv.org/abs/nlin/0310012 | |
| dc.identifier | http://arxiv.org/abs/nlin/0310012 | |
| dc.identifier | Acta Appl.Math. 83 (2004) 183-197 | |
| dc.identifier | doi:10.1023/B:ACAP.0000035597.06308.8a | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/80577 | |
| dc.subject | Exactly Solvable and Integrable Systems | |
| dc.subject | High Energy Physics - Theory | |
| dc.subject | Mathematical Physics | |
| dc.subject | Differential Geometry | |
| dc.subject | Symplectic Geometry | |
| dc.title | A simple way of making a Hamiltonian system into a bi-Hamiltonian one | |
| dc.type | text |