Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier

Francisco  Crespo; Francisco Javier  Molero; Sebastián  Ferrer

doi:10.3934/jgm.2016002

Article Contents

2016, Volume 8, Issue 2: 169-178. Doi: 10.3934/jgm.2016002

This issue Previous Article Invariant nonholonomic Riemannian structures on three-dimensional Lie groups Next Article Picard group of isotropic realizations of twisted Poisson manifolds

Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier

1.
Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C. Concepción
2.
Departamento de Matemática Aplicada, Universidad de Murcia, 30071 Espinardo

Received: August 2015

Revised: January 2016

Published: June 2016

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Abstract

Poisson and integrable systems are orbitally equivalent through the Nambu bracket. Namely, we show that every completely integrable system of differential equations may be expressed into the Poisson-Hamiltonian formalism by means of the Nambu-Hamilton equations of motion and a reparametrisation related by the Jacobian multiplier. The equations of motion provide a natural way for finding the Jacobian multiplier. As a consequence, we partially give an alternative proof of a recent theorem in [13]. We complete this work presenting some features associated to Hamiltonian maximally superintegrable systems.

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Mathematics Subject Classification: Primary: 34A26, 34A34; Secondary: 34A30, 70H05.

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