Morse index and linear stability of relative equilibria in singular mechanical systems

We have focussed on the study of the linear stability of some particular periodic orbits (called relative equilibria) in planar singular mechanical systems with SO(2)-symmetry, and we have achieved the results using quite advanced mathematical techniques. These involve some homotopy invariants, such as the spectral flow, and some index theory, namely a theorem stating the equality between the Morse index of an orbit seen as a critical point of a Lagrange action functional and the Maslov index of the fundamental solution of the associated Hamiltonian system. Moreover, what we have found meets one of its most important applications in a generalised n-body problem, that is, an n-body problem with a more general potential.