Equivariant Morse theory for starshaped Hamiltonian systems

Let $\Sigma$ be a starshaped hypersurface in ${R^{2n}}$; the problem of finding closed characteristics of $\Sigma$ can be classically reduced to a variational problem. This leads to studying an ${S^1}$-equivariant functional on a Hilbert space. The equivariant Morse theory of this functional, together with the assumption that $\Sigma$ only has finitely many geometrically distinct characteristics, leads to a remarkable formula relating the average indices of the characteristics. Using this formula one can prove, at least for $n$ even, that genetically there are infinitely many characteristics (cf. [E1] for the convex case).