A Case Study of Separability of Second-Order Differential Equations

Abstract

We perform a systematic study and classification of the possibility to decouple a class of second-order differential equations which have quadratic terms, either in the velocities or in the coordinates, plus linear damping terms. Our basic approach is the existing coordinate-free theory for the characterization of separable systems. But the focus on this specific class of second-order equations spontaneously leads to interesting side issues, such as the question of linearizability of the equations, and a potential role for other, less geometrical ideas of decoupling, such as the so-called technique of phase synchronization for linear systems.