The central topic of this thesis is the study of persistence of stationnary motion under explicit symmetry breaking perturbations in Hamiltonian systems. Explicit symmetry breaking occurs when a dynamical system having a certain symmetry group is perturbed in a way that the perturbation preserves only some symmetries of the original system. We give a geometric approach to study this phenomenon in the setting of equivariant Hamiltonian systems. A lower bound for the number of orbits of equilibria and orbits of relative equilibria which persist after a small perturbation is given. This bound is given in terms of the equivariant Lyusternik-Schnirelmann category of the group orbit. We also find a localization formula for this category in terms of the closed orbit-type strata. We show that this formula holds for topological spaces admitting a particular cover, made of tubular neighbourhoods of their minimal orbit-type strata. Finally we propose a construction of symplectic slices for subgroup actions.