Introduction to Nambu's Generalized Hamiltonian Dynamics

This book introduces Nambu's generalized Hamiltonian dynamics. In 1973, Nambu proposed extending classical Hamiltonian mechanics by replacing the canonical doublet ( p , q ) with a three-dimensional phase space defined by a canonical triplet ( x , y , z ). The equations of motion are formulated using a triple bracket-a generalization of the Poisson bracket-with two 'Hamiltonians' treated on an equal footing. This framework can further be extended to an n -tuple of phase-space coordinates, an n -bracket, and equations of motion involving n -1 Hamiltonians in an n -dimensional phase space. Nambu's original motivation was to generalize the Liouville theorem, which states that the volume of an ensemble in phase space is preserved under dynamical flows-a principle fundamental to statistical mechanics. He sought to construct systems with analogous properties for arbitrary-dimensional phase spaces, including odd dimensions. Although his proposal attracted little attention for more than a decade, subsequent developments revealed its relevance in diverse areas of theoretical and mathematical physics, notably in string/M-theory and fluid mechanics. This book introduces the reader to classical Nambu dynamics by explaining its principal aspects from an elementary viewpoint and developing it further from a coherent and unified standpoint. It is intended for readers with a reasonable understanding of classical analytical mechanics and working knowledge of basic physics and standard mathematical methods in theoretical physics.