On a unified formulation of completely integrable systems

TL;DR

This paper demonstrates that certain completely integrable systems with multiple conservation laws can be locally transformed into a simple linear form, with applications shown in biological and mechanical dynamical systems.

Contribution

It establishes a unified framework for representing integrable systems as equivalent to linear systems under specific conditions.

Findings

Integrable systems with $n-1$ conservation laws are locally equivalent to linear systems.

The results apply to the Lotka-Volterra system and Euler equations of rigid body dynamics.

Provides a method to simplify the analysis of complex dynamical systems.

Abstract

The purpose of this article is to show that a $\mathcal{C}^1$ differential system on $\R^n$ which admits a set of $n-1$ independent $\mathcal{C}^2$ conservation laws defined on an open subset $\Omega\subseteq \R^n$, is essentially $\mathcal{C}^1$ equivalent on an open and dense subset of $\Omega$, with the linear differential system $u^\prime_1=u_1, \ u^\prime_2=u_2,..., \ u^\prime_n=u_n$. The main results are illustrated in the case of two concrete dynamical systems, namely the three dimensional Lotka-Volterra system, and respectively the Euler equations from the free rigid body dynamics.