In this chapter, we state the basic principles behind the symmetry reduction of mechanical systems. We show that if a mechanical system allows a Lie group symmetry, and the configuration manifold is the same Lie group, then the Euler-Lagrange equations can be written in terms of vector variables defined on the Lie algebra. This calculation connects our derivation of Euler’s equations for the rigid body with the variational principles of mechanics. We also hint at how to extend this computation, due to Poincaré, for an arbitrary Lie group.