I suppose the potential energy e(C) for any configuration of n points C on S^2 is defined as the sum of reciprocal distances among all pairs of points in C. I guess distance d(x,y) could be reasonably defined by using either the arclength along S^2, or else Euclidean distance in R^3. Then E(n) is the minimum of e(C) over all configurations C of size n. We could call it E_a(n) if arclength is used, and E_e(n) if Euclidean distance is used. (It's not clear to me whether these would have distinct energy-minimizing configurations, but E_e is more physically realistic.) I would be very surprised if E(n) is algebraic. Reason being the great irregularity of the minimizing configurations in the classic problem P(n) of finding the maximum over size-n configurations C on S^2 of the minimum distance over all pairs of points in C. Very possibly, whenever sufficiently large n is of the form 6(a^2 - ab + b^2) for a, b in Z+ (if my hasty calculation is right), the optimal configuration will have isometry group = the icosahedral group I, #(I) = 60 (or occasionally the full isometry group of the icosahedron = I x Z/2Z, #(I x Z/2Z) = 120). (E.g., there is a continuum of equally minimum configurations for n = 5, and the maximum symmetry (of a cube) is broken for n = 8.) The connection is that using the inverse-square force law to define potential energy -- and then minimizing that P.E. over configurations C -- is like using the L^2 norm on reciprocal distances: the same configuration(s) will give minimum P.E. But if instead the L^p norm is used (on the 1/d(x,y)'s), then the classic problem above is what you get when you let p -> oo. --Dan P.S. I believe Neil has done extensive simulations to find minimum P.E. in a variety of situations (using Euclidean distance rather than arclength). On 2012-10-14, at 8:36 PM, James Propp wrote:
For n a positive integer, let E(n) be the minimal potential energy for any configuration of n points on the unit sphere subject to inverse-square-law repulsion. Must E(n) be algebraic? (That's gotta be true, but I don't see a rigorous argument.). Are there algorithms for computing E(n) (in the sense of finding an algebraic equation that it satisfies, and indicating which real root of the equation is intended)? How fast are these algorithms, and relatedly, what bounds can be given for the degree of E(n)?
Presumably this question would be part of a subject called "real algebraic programming" or something like that, but I don't know of any such subject (though I'm aware of Tarski's decision procedure for determining when a system of algebraic equations and inequalities admits a real solution).