  This paper expounds the modern theory of symplectic reduction in
finite-dimensional Hamiltonian mechanics. This theory generalizes the
well-known connection between continuous symmetries and conserved quantities,
i.e. Noether's theorem. It also illustrates one of mechanics' grand themes:
exploiting a symmetry so as to reduce the number of variables needed to treat a
problem. The exposition emphasises how the theory provides insights about the
rotation group and the rigid body. The theory's device of quotienting a state
space also casts light on philosophical issues about whether two apparently
distinct but utterly indiscernible possibilities should be ruled to be one and
the same. These issues are illustrated using ``relationist'' mechanics.
