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Local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction

I. K. Kozlov

Abstract

We prove that any bi-Hamiltonian system $v = \left(\mathcal{A} + λ\mathcal{B}\right)dH_λ$ that is Hamiltonian with respect all Poisson brackets $\mathcal{A} + λ\mathcal{B}$ is locally bi-integrable in both the real smooth case, when all eigenvalues of the Poisson pencil $\mathcal{P} = \left\{\mathcal{A} + λ\mathcal{B}\right\}$ are real, and in the complex analytic case. A complete set of functions in bi-involution is constructed by extending the set of standard integrals, which consists of Casimir functions of Poisson brackets, eigenvalues of the Poisson pencil and Hamiltonians.

Local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction

Abstract

We prove that any bi-Hamiltonian system that is Hamiltonian with respect all Poisson brackets is locally bi-integrable in both the real smooth case, when all eigenvalues of the Poisson pencil are real, and in the complex analytic case. A complete set of functions in bi-involution is constructed by extending the set of standard integrals, which consists of Casimir functions of Poisson brackets, eigenvalues of the Poisson pencil and Hamiltonians.

Paper Structure

This paper contains 18 sections, 12 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Conventions and acknowledgements
  3. Linear algebra
  4. Jordan--Kronecker theorem and Kronecker canonical form
  5. Core and mantle subspaces
  6. Admissible subspaces
  7. Bi-Lagrangian subspaces
  8. Poisson pencils
  9. Constructing new Poisson pencils using Casimir functions
  10. Core distribution
  11. Bi-Poisson reduction
  12. Linear bi-Poisson reduction
  13. Caratheodory--Jacobi--Lie theorem for Poisson manifolds
  14. Bi-Poisson reduction theorem
  15. Main theorem
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\mathcal{P} = \left\{ \mathcal{A} + \lambda \mathcal{B}\right\}$ be a Poisson pencil on a real $C^{\infty}$-smooth or complex analytic manifold $M$. In the real case, we assume that all eigenvalues of $\mathcal{P}$ are real. If a vector field $v$ is bi-Hamiltonian w.r.t. $\mathcal{P}$, then it

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Lemma 1.1
  • proof : Proof of Lemma \ref{['L:FirstMainT']}
  • Example 1.1
  • Remark 1.1
  • Theorem 2.1: Jordan--Kronecker theorem
  • Definition 2.1
  • Definition 2.2
  • ...and 28 more