Periodic nonlinear sliding modes for two uniformly magnetized spheres

Abstract

A uniformly magnetized sphere slides without friction along the surface of a second, identical sphere that is held fixed in space, subject to the magnetic force and torque of the fixed sphere and the normal force. The free sphere has two stable equilibrium positions and two unstable equilibrium positions. Two small-amplitude oscillatory modes describe the sliding motion of the free sphere near each stable equilibrium, and an unstable oscillatory mode describes the motion near each unstable equilibrium. The three oscillatory modes remain periodic at finite amplitudes, one bifurcating into mixed modes and circumnavigating the free sphere at large energies. For small energies, the free sphere is confined to one of the two discontiguous domains, each surrounding a stable equilibrium position. At large energies, these domains merge and the free sphere may visit both positions. The critical energy at which these domains merge coincides with the cumulation point of an infinite cascade of mixed-mode bifurcations. These findings exploit the equivalence of the force and torque between two uniformly magnetized spheres and the force and torque between two equivalent point dipoles, and offer clues to the rich nonlinear dynamics of this system. Online MagPhyx visualizations illustrate the dynamics.

Periodic bouncing modes for two uniformly magnetized spheres. II. Scaling

“…3 In this study, we take advantage of our recent proof that simple dipolar interactions exactly describe the magnetic interactions between uniformly magnetized spheres 4 and build upon our subsequent studies of dynamical interactions between spheres that remain in contact, both with and without friction. 5,6 Our 1497 modes exhibit a wide variety of behaviors, including simple modes with small m and n and complicated modes with large m and n. Mode (157,580,2) is the most complicated that we discovered. With E mnp = −0.003 999 and T mnp = 11 298, this mode requires about 16 000 Runge-Kutta time steps to integrate.…”

Periodic bouncing modes for two uniformly magnetized spheres. I. Trajectories

“…We have found such periodic states when the free sphere is constrained to remain in frictionless contact with the fixed sphere at all times. 14 We expect such states to exist for bouncing modes, and we are interested in searching for them.…”

“…Given that the initial position of the free sphere is always the same, it is the values of the initial momenta p r (0), p θ (0), and p φ (0) that completely determine the subsequent motion of the sphere. Because E = −1/3 is a minimum and because states with E ≥ 0 are unbounded (meaning that the free sphere eventually drifts off to infinity), 14 we seek periodic states with −1/3 < E < 0. Finding a particular mode (m, n, p) is guaranteed only for m n, for which Eqs.…”

“…These assumptions give periodic nonlinear states and infinite cascades of mixed-mode bifurcations. 14 We have also studied the dynamics of two freely rotating uniformly magnetized spheres that are in frictionless contact with each other. 15 In this paper, we study the conservative dynamics of a free sphere that makes contact with a fixed sphere only during frictionless elastic hard-sphere collisions, and we, therefore, include orbital, spin, and radial degrees of freedom for the free sphere.…”

“…Details of our fourth-order Runge-Kutta integration of the equations of motion, our treatment of hard-sphere collisions, and our code validation by comparison with various analytical limits are discussed elsewhere (Ref. 14 We take advantage of our recent proof that simple dipolar interactions exactly describe the magnetic interactions between uniformly magnetized spheres. 13 In Paper II, we explore scaling relationships for the energy and period of periodic modes discussed in this paper.…”