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WOLFRAM NOTEBOOK
The Thomson Problem with Central Forces
The Thomson Problem with Central Forces
The Thomson problem concerns the equilibrium positions of identical point charges constrained to move, without friction, on the surface of a sphere. Instead of motion constrained to a sphere, we consider motion in a central field with varying exponents, produced by a spherically symmetric charge density inside the sphere, with a total charge opposite to that of the point particles. For the case of a constant charge density in the sphere, the default setting, this is exactly the configuration that Thomson studied as a model for atoms (the plum pudding model), which suggested the present problem. The equilibrium positions are found here by the relaxation method—that is, by friction-damped motion from random stationary initial positions of the particles. Remarkably, the particles find their final resting positions on a spherical surface, although a naive expectation might favor a configuration with differing radial positions.
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The main intention of this Demonstration is not to clarify the questions concerning the actual equilibrium positions for small numbers of particles, but rather to present efficient computational dynamics for an -particle system with conservative pairwise forces, velocity-dependent frictional forces, and a conservative external force field in 3D.
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