Canonical form of Nambu–Poisson bracket: A pedestrian approach

Abstract

In the seventies, Nambu (Phys. Rev. D7, 2405 (1973)) proposed a new approach to classical dynamics based on an N -dimensional Nambu-Poisson (NP) manifold replacing the primi- tive even-dimensional Poisson manifold and on N -1 Hamiltonians in place of a single Hamiltonian. This approach has had many promoters including Bayen and Flato (Phys. Rev. D11, 3049 (1975)), Mukunda and Sudarshan (Phys. Rev. D13, 2846 (1976)), and Takhtajan (Comm. Math. Phys. 160, 295 (1994)) among others. While Nambu had originally considered N = 3, the illustration of his ideas for N = 4a nd 6w as given by Chatterjee ( Lett. Math. Phys. 36, 117 (1996)) who observed that the classical description of dynamical systems having dynamical symmetries is described ele- gantly by Nambu's formalism of mechanics. However, his considerations do not quite yield the beautiful canonical form conjectured by Nambu himself for the N -ary NP bracket. By making a judi- cious choice for the 'extra constant of motion' of namely, α and β, which are the orientation angles in Kepler problem and isotropic harmonic oscillator (HO) respectively, we show that the dynami- cal systems with dynamical symmetries can be recast in the beautiful form suggested by Nambu. We believe that the techniques used and the theorems suggested by us in this work are of gen- eral interest because of their involvement in the transition from Hamiltonian mechanics to Nambu mechanics.