The aim of the talk is to present two versions of the Liapunov center theorem for symmetric potentials. Let \Omega\subset \R^n be an open and invariant subset of an orthogonal representation \R^n of a compact Lie group \Gamma with free \Gamma-action. If \Gamma(q_0) 
\subset \Omega \cap (U')^{-1}(0) is a non-degenerate (i.e. dim kerU''(q_0) = dim \Gamma(q0)) or minimal orbit of critical points of \Gamma-invariant C^2-potential U :\Omega -> \R and there is at least one positive eigenvalue of the Hessian U''(q_0) then in any
neighborhood of the orbit \Gamma(q_0) there is a periodic orbit of non-stationary solutions of equation q_{tt}(t) = -U'(q(t)): Moreover, we estimate the minimal period of these solutions.
The basic idea of the proof is to apply the innite-dimensional version of the equivariant Conley index theory.

Conferencista: Slawomir Rybicki, Nicolaus Copernicus University
Torun, Poland.