A new perspective on the Kosambi-Cartan-Chern theory, and its applications

TL;DR

This paper offers a new perspective on the Kosambi-Cartan-Chern (KCC) theory by applying it to first order dynamical systems, exploring geometric invariants and stability, with applications to Hamiltonian systems.

Contribution

It introduces an alternative approach to KCC theory for first order systems and analyzes stability properties, including a detailed study of two-dimensional autonomous systems.

Findings

Established relationship between linear and Jacobi stability.

Applied KCC formalism to Hamiltonian systems with one degree of freedom.

Provided geometric interpretation of dynamical system stability.

Abstract

A powerful mathematical method for the investigation of the properties of dynamical systems is represented by the Kosambi-Cartan-Chern (KCC) theory. In this approach the time evolution of a dynamical system is described in geometric terms, treating the solution curves of a dynamical system by geometrical methods inspired by the geodesics theory of Finsler spaces. In order to geometrize the dynamical evolution one introduces a non-linear and a Berwald type connection, respectively, and thus the properties of any dynamical system are described in terms of five geometrical invariants, with the second one giving the Jacobi stability of the system. The Jacobi (in)stability is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the {\it non-metric} setting. Usually, the KCC theory is formulated by reducing the dynamical evolution equations to a set of second order differential equations. In this paper we present an alternative view on the KCC theory, in which the theory is applied to a first order dynamical system. After introducing the general framework of the KCC theory, we investigate in detail the properties of the two dimensional autonomous dynamical systems. The relationship between the linear stability and the Jacobi stability is also established. As a physical application of the formalism we consider the geometrization of Hamiltonian systems with one degree of freedom, and their stability properties.