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This article is cited in 29 scientific papers (total in 29 papers)
Proceedings of GDIS 2008, Belgrade
Bi-Hamiltonian structures and singularities of integrable systems
A. V. Bolsinova, A. A. Oshemkovb a School of Mathematics, Loughborough University, Ashby Road, Loughborough, LE11 3TU, UK
b Department of Mathematics and Mechanics, M.V. Lomonosov Moscow State University, Moscow, 119899, Russia
Abstract:
A Hamiltonian system on a Poisson manifold $M$ is called integrable if it
possesses sufficiently many commuting first integrals $f_1, \dots f_s$
which are functionally independent on $M$ almost everywhere. We study
the structure of the singular set $K$ where the differentials $df_1,
\dots, df_s$ become linearly dependent and show that in the case of
bi-Hamiltonian systems this structure is closely related to the
properties of the corresponding pencil of compatible Poisson brackets.
The main goal of the paper is to illustrate this relationship and to show
that the bi-Hamiltonian approach can be extremely effective in the study
of singularities of integrable systems, especially in the case of many
degrees of freedom when using other methods leads to serious
computational problems. Since in many examples the underlying
bi-Hamiltonian structure has a natural algebraic interpretation, the
technology developed in this paper allows one to reformulate analytic and
topological questions related to the dynamics of a given system into pure
algebraic language, which leads to simple and natural answers.
Keywords:
integrable Hamiltonian systems, compatible Poisson structures, Lagrangian fibrations, bifurcations, semisimple Lie algebras.
Received: 19.05.2009 Accepted: 10.06.2009
Citation:
A. V. Bolsinov, A. A. Oshemkov, “Bi-Hamiltonian structures and singularities of integrable systems”, Regul. Chaotic Dyn., 14:4-5 (2009), 431–454
Linking options:
https://www.mathnet.ru/eng/rcd958 https://www.mathnet.ru/eng/rcd/v14/i4/p431
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