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Bi-Hamiltonian structures and singularities of integrable systems
journal contribution
posted on 2014-10-01, 13:05 authored by Alexey BolsinovAlexey Bolsinov, A.A. OshemkovHamiltonian 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 DYNAMICSVolume
14Issue
4-5Pages
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
2009Notes
This a preprint version of the paper accepted for publication in Regular and Chaotic Dynamics (© Pleiades Publishing, Ltd. 2009).ISSN
1560-3547eISSN
1468-4845Publisher version
Language
- en