Extremizers of repulsive potential This question is about whether one can characterize the extremizers of the Coulomb energy $E:(\mathbb S^{2})^N \rightarrow [0,\infty]$ for $N$ points on the sphere with $x=(x_1,...,x_N)$ where $x_i \in \mathbb S^{2}$
$$E(x):=\sum_{i \neq j} \frac{1}{\Vert x_i-x_j \Vert_{\mathbb R^3}}$$
where $E(x)=\infty$ if two points coincide. 
Is the set of critical points known? Obviously it is invariant under rotations, so we may discard rotational symmetry in this question.
 A: This is called the Thomson problem, which was posed by Thomson in 1904. Global minima are known only for a few very specific values of the number of charges $N$. For example:


*

*for $N=3$, the solution is the set of vertices of a triangle.

*for $N=4$, 6, or 12, the solution is the set of vertices of the corresponding Platonic solid (tetrahedron, octahedron, icosahedron).

*for $N=5$, the solution is the set of vertices of triangular dipyramid.

*for $N=8$ and for $N=20$, the solutions are not the vertices of the corresponding Platonic solids (cube and dodecahedron). These are local minima but not global minima. Intuitively, non-triangular faces leave too much empty space at their centers and are not optimal.

*and so on. The Wikipedia article includes a table of some "good" results for certain values of $N$. In most cases, these are just the best numerical results found to date, with no proof that they are global minima. Some more candidates for global minima for many values of $N$ are presented in [1], and the coordinates of the points can be downloaded from this website.
Gradient descent works well to find good (possibly local) minima for this problem [2]. There is extensive literature on the topic: you can have a look at the links I provided here and the references therein.
[1] David J. Wales and Sidika Ulker, "Structure and dynamics of spherical crystals characterized for the Thomson problem," Phys. Rev. B 74, 212101 – (2006)
[2] Gautam, Simanta and Dmitry Vaintrob. “A Novel Approach to the Spherical Codes Problem.” MIT (2013) Available online here.
