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Local bi-integrability of bi-Hamiltonian systems, Part II: Real smooth case

I. K. Kozlov

Abstract

We prove that any bi-Hamiltonian system $v = \left(\mathcal{A} + λ\mathcal{B}\right)dH_λ$ on a real smooth manifold that is Hamiltonian with respect all Poisson brackets $\left(\mathcal{A} + λ\mathcal{B}\right)$ is locally bi-integrable. We construct a complete set of functions $\mathcal{G}$ in bi-involution by extending the set of standard integrals $\mathcal{F}$ consisting of Casimir functions of Poisson brackets, eigenvalues of the Poisson pencil, and the Hamiltonians. Moreover, we show that at a generic point of $M$ differentials of the extended family $d \mathcal{G}$ can realize any bi-Lagrangian subspace $L$ containing the differentials of the standard integrals $d \mathcal{F}$.

Local bi-integrability of bi-Hamiltonian systems, Part II: Real smooth case

Abstract

We prove that any bi-Hamiltonian system on a real smooth manifold that is Hamiltonian with respect all Poisson brackets is locally bi-integrable. We construct a complete set of functions in bi-involution by extending the set of standard integrals consisting of Casimir functions of Poisson brackets, eigenvalues of the Poisson pencil, and the Hamiltonians. Moreover, we show that at a generic point of differentials of the extended family can realize any bi-Lagrangian subspace containing the differentials of the standard integrals .

Paper Structure

This paper contains 12 sections, 11 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Conventions and acknowledgements
  3. Bi-Lagrangian subspaces
  4. Poisson pencils
  5. Constructing new Poisson pencils using Casimir functions
  6. Poisson pencils with common Casimirs
  7. Bi-Poisson reduction
  8. Linear bi-Poisson reduction
  9. Bi-Poisson reduction theorem
  10. Factorization theorem
  11. Standard integrals
  12. Proof of Theorem \ref{['T:MainGenTh']}

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 manifold $M$. If a vector field $v$ is bi-Hamiltonian w.r.t. $\mathcal{P}$, then it is locally bi-integrable.

Theorems & Definitions (21)

  • Definition 1.1
  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.1: Kozlov24BiLagr
  • Corollary 2.1
  • proof : Proof of Assertion \ref{['A:BrackComCas']}
  • Definition 3.1
  • ...and 11 more