Explaining the singular oscillator: physical considerations and mechanical analogies

Abstract Nonlinear oscillations in one dimension are typically described by a general differential equation of the form x ″ + f ( x ) = 0. In many classical systems, oscillations occur around an equilibrium position, defined where f ( x ) = 0. A notable exception is the singular oscillator, where f ( x ) = c / x , with c as a positive constant. Despite extensive studies yielding approximate periodic solutions expressed as series of sinusoidal functions, the singularity at x = 0 prevents this point from being a true equilibrium position. This raises critical questions: How can periodic solutions of the singular oscillator be explained intuitively? Is it physically rational for oscillations to occur around a position where the restoring force is undefined? This paper addresses these questions using simple mechanical analogies, such as a bouncing ball and the motion of a small object’s image in a mirror, to provide intuitive explanations of the periodic solutions. This analysis is particularly useful for teaching nonlinear phenomena at the university level and will assist young scientists with limited experience in the field.