Singularities of Bi-Hamiltonian Systems

Abstract

We study the relationship between singularities of bi-Hamiltonian systems and algebraic properties of compatible Poisson brackets. As the main tool, we introduce the notion of linearization of a Poisson pencil. From the algebraic viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with a fixed 2-cocycle. In terms of such linearizations, we give a criterion for non-degeneracy of singular points of bi-Hamiltonian systems and describe their types.

The U(n) free rigid body: Integrability and stability analysis of the equilibria

“…Although completeness can be shown as a direct consequence of Theorem 2.7, we prefer a proof of the complete integrability for the U(n) free rigid body dynamics based on the Bolsinov-Oshemkov codimension two principle, first, because it is natural from the viewpoint of the bi-Hamiltonian structure of the U(n) free rigid body dynamics and, second, since the proof can be performed directly, without restricting the U(n) free rigid body dynamics to the level hyperplanes of I (1) 0 and then invoking the complete integrability of the SU(n) free rigid body (see, [26,27,14]). We emphasize that our proof of the complete integrability gives an application of the Bolsinov-Oshemkov method to a bi-Hamiltonian system on a non-semi-simple Lie algebra, a case that is not discussed in detail in [8,7]. At the end of the section, we also mention another proof of the complete integrability of the U(n) free rigid body that uses a theorem of Brailov on completely involutive sets of functions on affine Lie algebras; see [14, Chapter 5, §20.2] for a nice presentation of this result.…”

“…[35,12]). A more sophisticated description of the properties of the singularities of bi-Hamiltonian systems is given in [7]. We recall that all Hamiltonian systems on the cotangent bundle of a Lie group for a left-invariant Hamiltonian can be reduced to a Lie-Poisson system on the dual of the corresponding Lie algebra (see, e.g., [23,31]).…”

Finite-dimensional integrable systems: A collection of research problems

“…A recent series of papers [6,10,27] has been devoted to the study of singularities of bi-Hamiltonian systems. The underlying bi-Hamiltonian structure discussed in these papers is related to Poisson pencils P = {A 0 +λA 1 } of Kronecker type (see Section 4.1).…”

“…Could we use the technology developed in [10] to study this problem? The idea of such an alternative approach is as follows.…”

“…For example, even if we start with quite simple brackets (e.g., linear as in our example), the new reduced brackets are not linear any more which leads to many technical problems (e.g., we do not have any natural coordinate system for explicit computations). The main idea of [10] is to "replace" a (non-linear) Poisson pencil by its linearisation. For our purposes, this linearisation needs to be done at an equilibrium point of the corresponding reduced Hamiltonian system (more precisely, at a common equilibrium point for all reduced Hamiltonians).…”