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Periodic bouncing modes for two uniformly magnetized spheres. I. Trajectories
Boyd F Edwards1, Bo A Johnson1, John M Edwards2
1Department of Physics, Utah State University, Logan, Utah 84322, USA.
Chaos (Woodbury, N.Y.)
|February 5, 2020
Summary
Researchers studied the motion of two magnetized spheres, finding 1243 unique periodic collision paths. These paths exhibit complex behaviors and symmetric trajectories, offering insights into nonlinear dynamics.
Area of Science:
- Physics
- Nonlinear Dynamics
- Magnetohydrodynamics
Background:
- Investigates the dynamics of interacting magnetized spheres.
- Focuses on elastic collisions and periodic motion in a frictionless environment.
Purpose of the Study:
- To find and characterize periodic solutions for the nonlinear equations of motion governing a free sphere interacting with a fixed sphere.
- To explore the behavior of large-amplitude modes and their relationship to small-amplitude solutions.
Main Methods:
- Employs Runge-Kutta integration for numerical analysis of large-amplitude modes.
- Derives closed-form mathematical solutions for small-amplitude modes to validate numerical findings.
Main Results:
- Identified 1243 distinct periodic modes originating from the stable equilibrium position.
- Characterized bifurcations from radial bouncing modes to states with significant angular motion.
- Observed complex trajectories with numerous collisions and angular oscillations per period.
Conclusions:
- The system exhibits a rich variety of behaviors and symmetric trajectories.
- Periodic solutions are abundant and demonstrate complex dynamics beyond simple radial motion.
- The study validates numerical findings with analytical solutions for small-amplitude oscillations.
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