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Title: Bi-Hamiltonian structures and singularities of integrable systems
Authors: Bolsinov, Alexey V.
Oshemkov, A.A.
Keywords: Integrable Hamiltonian systems
Compatible Poisson structures
Lagrangian fibrations
Bifurcations
Semisimple Lie algebras
Issue Date: 2009
Publisher: SP MAIK Nauka/Interperiodica/Springer (© Pleiades Publishing, Ltd.)
Citation: BOLSINOV, A.V. and OSHEMKOV, 2009. Bi-Hamiltonian structures and singularities of integrable systems. Regular and Chaotic Dynamics, 14 (4-5), pp.431-454.
Abstract: Hamiltonian system on a Poisson manifold M is called integrable if it possesses sufficiently many commuting first integrals f1, . . . fs which are functionally independent on M almost everywhere. We study the structure of the singular set K where the differentials df1, . . . , dfs 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.
Description: This a preprint version of the paper accepted for publication in Regular and Chaotic Dynamics (© Pleiades Publishing, Ltd. 2009).
Version: Submitted for publication
DOI: 10.1134/S1560354709040029
URI: https://dspace.lboro.ac.uk/2134/15958
Publisher Link: http://dx.doi.org/10.1134/S1560354709040029
ISSN: 1560-3547
Appears in Collections:Published Articles (Maths)

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