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Integrability of bi-Hamiltonian systems using Casimir functions and characteristic polynomials

I. K. Kozlov

Abstract

In this paper we prove that for a pencil of compatible Poisson brackets $\mathcal{P} = \left\{\mathcal{A} + λ\mathcal{B} \right\}$ the local Casimir functions of Poisson brackets $\mathcal{A} + λ\mathcal{B}$ and coefficients of the characteristic polynomial $p_{\mathcal{P}}$ commute w.r.t. all Poisson brackets of the pencil $\mathcal{P}$. We give a criterion when this family of functions is complete. These results generalize previous constructions of complete commutative subalgebras in the symmetric algebra $S(\mathfrak{g})$ of a finite-dimensional Lie algebra $\mathfrak{g}$ by A.S. Mishchenko & A.T. Fomenko, A.V. Bolsinov & P. Zhang and A.M. Izosimov.

Integrability of bi-Hamiltonian systems using Casimir functions and characteristic polynomials

Abstract

In this paper we prove that for a pencil of compatible Poisson brackets the local Casimir functions of Poisson brackets and coefficients of the characteristic polynomial commute w.r.t. all Poisson brackets of the pencil . We give a criterion when this family of functions is complete. These results generalize previous constructions of complete commutative subalgebras in the symmetric algebra of a finite-dimensional Lie algebra by A.S. Mishchenko & A.T. Fomenko, A.V. Bolsinov & P. Zhang and A.M. Izosimov.

Paper Structure

This paper contains 18 sections, 25 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Jordan--Kronecker theorem
  4. Real Jordan--Kronecker theorem
  5. Characteristic polynomial
  6. Core and mantle subspaces
  7. Poisson pencils
  8. Core distribution
  9. Local coordinates for core and mantle
  10. Factorization theorem
  11. Eigenvalues of Poisson pencils
  12. Complex conjugate eigenvalues
  13. Main result
  14. Extended core distribution
  15. Completeness criterion
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $A$ and $B$ be skew-symmetric bilinear forms on a finite-dimension vector space $V$ over a field $\mathbb{K}$ with $\textmd{char } \mathbb{K} =0$. If the field $\mathbb{K}$ is algebraically closed, then there exists a basis of the space $V$ such that the matrices of both forms $A$ and $B$ are b where each pair of corresponding blocks $A_i$ and $B_i$ is one of the following:

Theorems & Definitions (47)

  • Theorem 2.1: Jordan--Kronecker theorem
  • Theorem 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Definition 2.8
  • Proposition 2.9
  • Proposition 2.10
  • ...and 37 more