Extremals of the Dirichlet functional on the manifold of two-dimensional spheroids in SO(3)

An approach to the approximate calculation of extremals of the Dirichlet functional on the Banach Lie group of “singular spheroids” defined in an annulus on the coordinate plane centered at the origin is presented. The technique presented is based on the use of a special simplification (functional reduction) of the equation of extremals. It is shown that the functional reduction allows one to obtain a lower estimate for the Morse index of extremals. We use smooth Fredholm functionals with continuous symmetries on smooth Banach manifolds with Riemannian (Hilbert) framings.