Nambu Brackets With Constraint Functionals

Abstract

If a Hamiltonian dynamical system with n degrees of freedom admits m constants of motion more than 2n − 1, then there exist some functional relations between the constants of motion. Among these relations the number of functionally independent ones are s = m − (2n − 1). It is shown that for such a system in which the constants of motion constitute a polynomial algebra closing in Poisson bracket, the Nambu brackets can be written in terms of these s constraint functionals. The exemplification is very rich and several of them are analyzed in the text. IntroductionThe concept of generalized Hamiltonian dynamics arouse in 1973 with an article by Y. Nambu.[1] In his proposal, Nambu employed an N -ary bracket, generically called Nambu bracket (NB), to describe the time evolution of the dynamical system in N -dimensional (N D) phase space. His bracket includes N − 1 functionally independent constants of motion, the so-called generalized Hamiltonians. As an illustrative example, Nambu considered the Euler equations of free rigid body for a 3D phase space and this was the only example given. Finding examples in higher odd-dimensions is still very tedious matter.In Nambu formalism, dynamical systems produce inevitably a nontrivial normalization factor C at least when N is an even integer grater than three. [2] In words, in order to get the correct Hamiltonian dynamics the NBs must be normalized properly. The explicit general form of C has been derived in detail for superintegrable systems.[3] The aim of this paper is to obtain C for a Hamiltonian system with n degrees of freedom and m constants of motion C 1 , · · · , C m , with m ≥ 2n. defines NB of N -th order satisfying the properties skew-symmetry, Leibniz rule and generalized Jacobi identity (fundamental identity).[4] When one considers this NB structure C ∞ (M) admits another algebra structure.A. Tegmen Nambu brackets with constraint functionals 2