Using the fundamental equation of constrained motion with inequality constraints

In this paper we investigate a computational approach to keeping a moving particle within a predefined annulus or a predefined bounded space, formed by two concentric spheres with radii Lmin and Lmax, respectively, assuming that said particle cannot maintain a perfectly circular trajectory. The study develops an algorithm for dealing with a system in which constraints are expressed as inequalities. The proposed approach expresses the trajectory in terms of winding/unwinding logarithmic spirals with transitions, expressed as damped vibrations, between them. These transitions are necessary to resolve incompatibility between initial conditions for winding/unwinding spirals. Equations of motion for the particle are obtained by using the Fundamental Equation of Constrained Motion. The obtained simulation results show that such an approach produces the desired pseudo-periodic type of motion, and the particle stays within the predefined region of space for a long duration, although no cycle of its trajectory is repeated.