|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'504'928
Articles rated: 2609
25 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
A simple way of making a Hamiltonian system into a bi-Hamiltonian one | | A. Sergyeyev
; | | Date: |
13 Oct 2003 | | Journal: | Acta Appl.Math. 83 (2004) 183-197 DOI: 10.1023/B:ACAP.0000035597.06308.8a | | Subject: | Exactly Solvable and Integrable Systems; Symplectic Geometry; Mathematical Physics; Differential Geometry | nlin.SI hep-th math-ph math.DG math.MP math.SG | | Abstract: | Given a Poisson structure (or, equivalently, a Hamiltonian operator) $P$, we show that its Lie derivative $L_{ au}(P)$ along a vector field $ au$ defines another Poisson structure, which is automatically compatible with $P$, if and only if $[L_{ au}^2(P),P]=0$, where $[cdot,cdot]$ is the Schouten bracket. We further prove that if $dimker Pleq 1$ 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 $L_{ au}(P)$, where $ au$ is a local vector field such that $L_{ au}^2(P)=L_{ ilde au}(P)$ for some other local vector field $ ilde au$. 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 | | Source: | arXiv, nlin.SI/0310012 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER