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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 infinitesimal-amplitude 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.

A vibrant online learning community shares information about building beautiful sculptures from collections of small neodymium magnet spheres, with YouTube tutorial videos attracting over a hundred million views.1,2 These spheres offer engaging hands-on exposure to principles of magnetism and are used both in and out of the classroom to teach principles of mathematics, physics, chemistry, biology, and engineering.3 We showed recently that the forces and torques between two uniformly magnetized spheres are identical to the forces and torques between two point magnetic dipoles. In this paper, we exploit this equivalence to study the conservative nonlinear dynamics of a uniformly magnetized sphere subject to the magnetic forces and torques produced by a second, fixed, uniformly magnetized sphere, assuming frictionless hard-sphere elastic collisions between them. Our search for periodic states uncovers a wide variety of periodic modes, some of which are highly complex and beautiful.

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