  In 1956, Bott in his celebrated paper on closed geodesics and Sturm
intersection theory, proved an Index Iteration Formula for closed geodesics on
Riemannian manifolds. Some years later, Ekeland improved this formula in the
case of convex Hamiltonians and, in 1999, Long generalized the Bott iteration
formula by putting in its natural symplectic context and constructing a very
effective Index Theory. The literature about this formula is quite broad and
the dynamical implications in the Hamiltonian world (e.g. existence,
multiplicity, linear stability etc.) are enormous.
  Motivated by the recent discoveries on the stability properties of symmetric
periodic solutions of singular Lagrangian systems, we establish a Bott-type
iteration formula for dihedrally equivariant Lagrangian and Hamiltonian
systems.
  We finally apply our theory for computing the Morse indices of the celebrated
Chenciner and Montgomery figure-eight orbit for the planar three body problem
in different equivariant spaces. Our last dynamical consequence is an
hyperbolicity criterion for reversible Lagrangian systems.
