Jacobi Stability of Linearized Geometric Dynamics

Abstract

This paper is dedicated to the study of geometric dynamics (an Euler-Lagrange prolongation of a flow on a Riemannian manifold) from the point of view of KCC theory, Jacobi stability and Lyapounov stability. Section 1 recalls the geometrical roots of Jacobi stability and announces the subject of the paper. Section 2 introduces the variational ODEs (Jacobi fields ODEs), the KCC differential invariants for a second order ODE system, and defines the Jacobi stability. Section 3 studies the KCC differential invariants associated to geometric dynamics. Section 4 describes various linearizations of geometric dynamics. Section 5 studies the Jacobi stability for linearized geometric dynamics around a stationary point of the field. Section 6 shows that the linearized geometric dynamics around a critical point of the energy can be Jacobi stable or unstable. Section 7 proves the Lyapounov instability of Jacobi fields ODEs along geometric dynamics trajectories.