# American Institute of Mathematical Sciences

June  2016, 8(2): 169-178. doi: 10.3934/jgm.2016002

## 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, Chile 2 Departamento de Matemática Aplicada, Universidad de Murcia, 30071 Espinardo, Spain

Received  August 2015 Revised  January 2016 Published  June 2016

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.
Citation: Francisco Crespo, Francisco Javier Molero, Sebastián Ferrer. Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier. Journal of Geometric Mechanics, 2016, 8 (2) : 169-178. doi: 10.3934/jgm.2016002
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