We will report on recent work in the area of stability/instability for families of periodic solutions determined by variational principles. The focus of this talk will be on applications to Hamiltonian systems with symmetry groups, and the instability of symmetric action minimizing orbits. We shall discuss several examples in the area of celestial mechanics, where the symmetry groups arise from assumptions made on the symmetry class of the periodic orbit. The figure eight orbit of Chenciner and Montgomery for the planar three body problem is an example, and will be briefly described. Another orbit family recently discovered by Chenciner-Venturelli , the Hip-Hop family, for the spatial four body problem, may be analyzed with the techniqes used to prove our results. We shall show why this family is generically hyperbolic on a reduced space (modulo symmetries). The numerical challenges of finding the orbits, and providing a stability analysis is thereby shown to be solved simultaneously, if the variational principle can be implemented numerically.