Lagrangian structures, integrability and chaos for 3D dynamical equations

TL;DR

This paper develops a Lagrangian framework for 3D autonomous dynamical systems, including integrable and chaotic cases, and introduces methods to identify constants of motion and analyze system properties.

Contribution

It presents a novel approach to construct Action Principles for 3D systems, linking Lagrangian and Hamiltonian structures to study integrability and chaos.

Findings

Lagrangian descriptions valid for systems with strange attractors

Method to find new constants of motion from known ones

Complete integrability of the ABC system under certain conditions

Abstract

In this paper we consider the general setting for constructing Action Principles for three-dimensional first order autonomous equations. We present the results for some integrable and non-integrable cases of the Lotka-Volterra equation, and we show Lagrangian descriptions which are valid for systems satisfying Shil'nikov criteria on the existence of strange attractors, though chaotic behavior or homoclinic orbits have not been verified up to now. The Euler-Lagrange equations we get for these systems usually present "time reparameterization" symmetry, though other kinds of invariance may be found according to the kernel of the associated symplectic 2-form. The formulation of a Hamiltonian structure (Poisson brackets and Hamiltonians) for these systems from the Lagrangian viewpoint leads to a method of finding new constants of the motion starting from known ones, which is applied to some systems found in the literature known to possess a constant of the motion, to find the other and thus showing their integrability. In particular, we show that the so-called ABC system is completely integrable if it possesses one constant of the motion. Pacs numbers: 45.20.Jj, 02.30.Ik, 05.45.Ac Keywords: Lagrangian, integrable system, chaos, Lotka Volterra