Vector Hamiltonians in Nambu mechanics

TL;DR

This paper generalizes Nambu mechanics using vector Hamiltonians, showing divergence-free flows in any dimension can be represented with multiple invariants, and introduces Cartan mechanics for flows with fewer invariants.

Contribution

It extends Nambu mechanics to higher dimensions with vector Hamiltonians and introduces Cartan mechanics for flows with limited invariants.

Findings

Any divergence-free flow in R^n can be represented as generalized Nambu mechanics.

Introduces Cartan mechanics for flows with n-3 or fewer invariants.

Provides the fifth integral invariant of Euler top as an example.

Abstract

We give a generalization of the Nambu mechanics based on vector Hamiltonians theory. It is shown that any divergence-free phase flow in $\mathbb{R}^n$ can be represented as a generalized Nambu mechanics with $n-1$ integral invariants. For the case when the phase flow in $\mathbb{R}^n$ has $n-3$ or less first integrals, we introduce the Cartan concept of mechanics. As an example we give the fifth integral invariant of Euler top.