In the univariate case the zeros of Jacobi-polynomials give Gauss-Jacobi quadratures which are distinguished by highest possible degree of exactness. Moreover, these quadratures have positive weights from which convergence of these formulae is guaranteed.
To advance to the multivariate case, an interpretation of the zeros of the Jacobi polynomials is helpful. The zeros of these orthogonal polynomials are closely connected to movable charges situated on a metal rod which is represented by the real interval . In the state of equilibrium the potential energy of the charge distribution is minimized and their locations coincide with the zeros of the Jacobi polynomials. For more details see Stroud and Secrest [24, p. 17,].
Thus a minimization of the potential energy of a point distribution on the sphere is a good criterion for an appropriate node distribution.
With the Euclidian norm in we get for
electrical charged point particles at locations
the potential energy
and thus the nonlinear optimization problem
which is a facility dispersion problem on the sphere. In computational physics and chemistry, this problem is also known as Thomson's problem.