On Yudin's scheme of finding a lower estimate for the potential energy of a system of charges on a sphere

We consider Yudin's problem of finding a lower estimate for the minimum of the potential energy of a system of equal charges on the unit sphere $\mathbb S^2$ of Euclidean space $\mathbb R^3$. For this problem, we write a dual problem in the case of an arbitrary number of charges. A solution of the dual problem is given for the cases of 5, 6, 7, 8, and 12 charges. In addition, the primal problem is solved and hypothetically optimal distributions are specified for 7 and 8 charges. It is established that Yudin’s method does not allow one to prove the optimality of these distributions.