Generalized Hamilton-Jacobi theory of Nambu Mechanics

TL;DR

This paper develops a Hamilton-Jacobi-like formulation for Nambu mechanics, extending classical mechanics to higher-dimensional phase spaces with multiple Hamiltonians, aiming to facilitate quantization.

Contribution

It introduces a novel Hamilton-Jacobi framework for Nambu mechanics, bridging a gap in the theoretical understanding and potential quantization methods.

Findings

Formulated a Hamilton-Jacobi-like equation for Nambu mechanics.

Extended classical mechanics to multi-Hamiltonian systems in higher dimensions.

Provided a new perspective for quantization of Nambu systems.

Abstract

We develop a Hamilton-Jacobi-like formulation of Nambu mechanics. The Nambu mechanics, originally proposed by Nambu more than four decades ago, provides a remarkable extension of the standard Hamilton equations of motion in even dimensional phase space with a single Hamiltonian to a phase space of three (and more generally, arbitrary) dimensions with two Hamiltonians (n Hamiltonians in the case of (n+1)-dimensional phase space) from the viewpoint of the Liouville theorem. However, it has not been formulated seriously in the spirit of Hamilton-Jacobi theory. The present study is motivated to suggest a possible direction towards quantization from a new perspective.