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In this note we will consider reduction techniques for Hamiltonian systemsthat are invariant under the action of a compact Lie group $G$ acting bysymplectic diffeomorphisms, and the related work on stability of relativeequilibria. We will focus on reduction by invariants in which case it ispossible to describe a reduced phase space within the orbit space byconstructing an orbit map using a Hilbert basis of invariants for the symmetrygroup $G$. Results considering the stratification, foliation and fibration ofthe phase space and the orbit space are considered. Finally some remarks aremade concerning relative equilibria and bifurcations of periodic solutions. Wewill combine results from a wide variety of papers. We obtain that for theorbit space the orbit type stratification coincides with the Thom-Boardmanstratification and each stratum is foliated with symplectic leaves. Furthermorethe orbit space is fibred into reduced phase spaces and each reduced phasespace has a orbit type stratification, where each orbit type stratum is asymplectic leaf.

Reduction by invariants, stratifications, foliations, fibrations and relative equilibria, a short survey