In this chapter, we extend the theory of reduction of Hamiltonian systems with symmetry to include systems with a discrete symmetry group acting symplectically. The exposition here is based on the work of Harnad, Hurtubise and Marsden [1991].

For antisymplectic symmetries such as reversibility, this question has been considered by Meyer [1981] and Wan [1990]. However, in this chapter we are concerned with symplectic symmetries. Antisymplectic symmetries are typified by time reversal symmetry, while symplectic symmetries are typified by spatial discrete symmetries of systems like reflection symmetry. Often these are obtained by taking the cotangent lift of a discrete symmetry of configuration space.

There are two main motivations for the study of discrete symmetries. The first is the theory of bifurcation of relative equilibria in mechanical systems with symmetry. The rotating liquid drop is a system with a symmetric relative equilibrium that bifurcates via a discrete symmetry. An initially circular drop (with symmetry group S1) that is rotating rigidly in the plane with constant angular velocity Ω, radius r, and with surface tension τ, is stable if r3Ω2 < 12τ (this is proved by the energy-Casimir or energy-momentum method). Another relative equilibrium (a rigidly rotating solution in this example) branches from this circular solution at the critical point r3Ω2 = 12τ. The new solution has the spatial symmetry of an ellipse; that is, it has the symmetry ℤ2 × ℤ2 (or equivalently, the dihedral group D2).