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Bi-Hamiltonian structures and singularities of integrable systems

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journal contribution
posted on 01.10.2014 by Alexey Bolsinov, A.A. Oshemkov
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.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

REGULAR & CHAOTIC DYNAMICS

Volume

14

Issue

4-5

Pages

431 - 454 (24)

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.

Publisher

SP MAIK Nauka/Interperiodica/Springer (© Pleiades Publishing, Ltd.)

Version

SMUR (Submitted Manuscript Under Review)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2009

Notes

This a preprint version of the paper accepted for publication in Regular and Chaotic Dynamics (© Pleiades Publishing, Ltd. 2009).

ISSN

1560-3547

eISSN

1468-4845

Language

en

Location

Belgrade, SERBIA

Exports