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

Abstract

We consider a uniformly magnetized sphere that moves without friction in a plane in response to the field of a second, identical, fixed sphere, making elastic hard-sphere collisions with this sphere. We seek periodic solutions to the associated nonlinear equations of motion. We find closed-form mathematical solutions for small-amplitude modes and use these to characterize and validate our large-amplitude modes, which we find numerically. Our Runge-Kutta integration approach allows us to find 1243 distinct periodic modes with the free sphere located initially at its stable equilibrium position. Each of these modes bifurcates from the finite-amplitude radial bouncing mode with infinitesimalamplitude angular motion and supports a family of states with increasing amounts of angular motion. These states offer a rich variety of behaviors and beautiful, symmetric trajectories, including states with up to 157 collisions and 580 angular oscillations per period.Published under license by AIP Publishing. https://doi.panels (a)-(d). The length scales for all four trajectories are the same and are set by the size of the gray (fixed) sphere.

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

“…The former scaling relationship applies only to threshold energies E mnp , that is, the minimum energy for mode (m, n, p), and the associated period T mnp . As the energy E increases beyond this minimum for a particular mode, the period T increases for some modes [such as (2,1,2), (2,3,2), and (3,1,2)] and decreases for others [such as (1,1,2), (1,2,2), (2,5,2)].…”

“…2 point from duplicate modes to the associated irreducible modes of the same energy. 1 For example, the dashed horizontal line at E = −0.065 starts the irreducible mode (1, 1, 2) and passes through the duplicate modes (2, 2, 2), (3, 3, 2), etc., of the same energy. The periods of these duplicate modes…”

“…As discussed in Ref. 1, the basis for all of these modes is the finite-amplitude radial mode. In this mode, the magnetic moments of the two spheres remain in the same direction as the motion of the free sphere-there is no angular motion as the south pole of the free sphere bounces repeatedly against the north pole of the fixed sphere.…”

“…In Paper I, 1 we use MagPhyx software to find 1497 periodic modes for a uniformly magnetized sphere moving without friction in response to the field of a second, identical, fixed sphere and making elastic hard-sphere collisions with this sphere. 2 Three quantities (m, n, p) characterize these modes: (1) the number m of collisions during one period of the motion, called the bouncing number, (2) the number n of angular oscillations during one period of the motion, called the rocking number, and (3) the initial phase of the angular oscillations, with p = 1 for the in-phase motion with initial orbital and spin angular momenta of the same sign and with p = 2 for the out-of-phase motion with initial orbital and spin angular momenta of opposite signs.…”

“…Selected mode descriptors (m, n, p) are shown to the left of the corresponding data points. Dashed horizontal lines point from duplicate modes to the associated irreducible modes of the same energy 1.…”