Nonlinear damped oscillators on Riemannian manifolds: Fundamentals

The classical theory of mass-spring-damper-type dynamical systems on the ordinary flat space R^3 may be generalized to higher-dimensional Riemannian manifolds by reformulating the basic underlying physical principles through differential geometry. Nonlinear dynamical systems have been studied in the scientific literature because they arise naturally from the modeling of complex physical structures and because such dynamical systems constitute the basis for several modern applications such as the secure transmission of information. The flows of nonlinear dynamical systems may evolve over time in complex, non-repeating (although deterministic) patterns. The focus of the present paper is on formulating the general equations that describe the dynamics of a point-wise particle sliding on a Riemannian manifold in a coordinate-free manner. The paper shows how the equations particularize in the case of some manifolds of interest in the scientific literature, such as the Stiefel manifold and the manifold of symmetric positive-definite matrices.