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

2024/10/29 by I. K. Kozlov, Kozlov, I. K.
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Numerical methods for differential equations #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2410.21642

Abstract

We prove that any bi-Hamiltonian system v = (A + λB)dH_λ on a real smooth manifold that is Hamiltonian with respect all Poisson brackets (A + λB) is locally bi-integrable. We construct a complete set of functions G in bi-involution by extending the set of standard integrals 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 G can realize any bi-Lagrangian subspace L containing the differentials of the standard integrals d F.

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