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.

References in corpus (2)

• Stability in bi-Hamiltonian systems and multidimensional rigid body
• Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain

Cited by in corpus (4)

• Nijenhuis Geometry
• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems
• Cotangent models for integrable systems
• Some Remarks on the Topology of Hyperbolic Actions of ${\mathbb{R}}^n$ on $n$-manifolds