The Thomson Problem with Central Forces

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.


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.

Apart from the obvious ones, the control labels are explained in tooltips that appear when you mouse over them, unless the "run" checkbox is checked.


Contributed by: Ulrich Mutze (September 2015)
Open content licensed under CC BY-NC-SA



Snapshot 1: an initial random configuration of particles

Snapshot 2: the final configuration after having come to rest in the default force field

Snapshot 3: the final configuration for a force field made by a positive space charge concentrated near the surface of the sphere

Snapshot 4: final configuration for a large number of particles

Snapshot 5: view with the sphere made transparent

Snapshot 6: view with the polyhedron removed from the particles

Snapshot 7: a non-default configuration not yet having come to rest

This Demonstration owes many coding details to the Demonstration "Repulsionhedra and the Thomson Problem" by Brian Burns.

The nature of energy minima at the equilibrium positions is studied in the Demonstration "Visualizing the Thomson Problem" by Mark Peterson.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.