Singularities of bihamiltonian systems

Details Singularities of bihamiltonian systems Singularities of bihamiltonian systems

2012-03-15

arxiv.org/abs/1203.3419v1

Physics

Mathematical Physics

Scientific paper

Two Poisson brackets are called compatible if any linear combination of these brackets is a Poisson bracket again. The set of non-zero linear combinations of two compatible Poisson brackets is called a Poisson pencil. A system is called bihamiltonian (with respect to a given pencil) if it is hamiltonian with respect to any bracket of the pencil. The property of being bihamiltonian is closely related to integrability. On the one hand, many integrable systems known from geometry and physics possess a bihamiltonian structure. On the other hand, if we have a bihamiltonian system, then the Casimir functions of the brackets of the pencil are integrals in involution of our system. We consider the situation when these integrals are enough for complete integrability of the system. As it was shown by Bolsinov and Oshemkov, many properties of the system in this case can be deduced from the properties of the Poisson pencil itself, without explicit analysis of integrals. Developing these ideas, we introduce the notion of linearization of a Poisson pencil. In terms of linearization, we give a criteria for non-degeneracy of a singular point and describe its type.

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