Nambu--Poisson reformulation of the finite dimensional dynamical systems

TL;DR

This paper introduces a Nambu--Poisson reformulation of finite-dimensional dynamical systems, demonstrating integrals of motion and solutions in specific cases, including a reduction to Volterra's system.

Contribution

It presents a novel reformulation of Hamiltonian dynamics using Nambu--Poisson structures and explicitly solves particular cases by quadratures.

Findings

Complete integrals of motion found in two cases

Reduction to Volterra's system in a special case

Systems solved explicitly by quadratures

Abstract

In this paper we introduce a system of nonlinear ordinary differential equations which in a particular case reduces to Volterra's system. We found in two simplest cases the complete sets of the integrals of motion using Nambu--Poisson reformulation of the Hamiltonian dynamics. In these cases we have solved the systems by quadratures.