Generalized Cartesian–Nambu Vector Fields

Jaume Llibre, Rafael Ramírez and Valentín Ramírez
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Jaume Llibre: Universitat Autònoma de Barcelona, Departament de Matemàtiques
Rafael Ramírez: Universitat Rovira i Virgili, Ingenieria Informatica i matemàtiques
Valentín Ramírez: Universitat Autònoma de Barcelona, Departament de Matemàtiques

Chapter Chapter 5 in Dynamics through First-Order Differential Equations in the Configuration Space, 2023, pp 177-283 from Springer

Abstract: Abstract Nambu mechanics is a generalization of Hamiltonian mechanics involving several Hamiltonians. Recall that Hamiltonian mechanics is based on the flows generated by a smooth Hamiltonian over a symplectic manifold. The flows are symplectomorphisms, i.e., a transformation of phase space that is volume preserving and preserves the symplectic structure of the phase space, and hence obeys Liouville’s Theorem. In 1973 Yoichiro Nambu suggested an extension of Hamiltonian dynamics, based on an N-dimensional Nambu–Poisson manifold replacing the even dimensional Poisson manifold, i.e., a manifold with a given Poisson bracket and replacing a single Hamiltonian H for N − 1 Hamiltonian H1, …, HN−1. In the canonical Hamiltonian formulation the equations of motion (Hamilton equations) are defined via the Poisson bracket.

Date: 2023
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DOI: 10.1007/978-3-031-27095-6_5

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