Comparison of finite-dimensional reductions in smooth variational problems with symmetries

We prove the symmetric version of the comparison theorem and show the global equivalence of the Lyapunov-Schmidt and Morse-Bott key functions for the potential of a natural mechanical system with symmetries.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 7, Suzdal Conference-1, 2003.This revised version was published online in April 2005 with a corrected cover date.