One of the principal ideas of geometric mechanics is reduction, which goes back to Jacobi's elimination of the node in the 3-body problem, but in its modern form is due to K. Meyer (1973) and independently J.E. Marsden and A. Weinstein (1974), both inspired by the work of Smale (1970). Symmetry of a Hamiltonian or Lagrangian system gives rise to conserved quantities, by Noether's theorem, and these conserved quantities are the components of the momentum map J. If P is the phase space and G the symmetry group, the momentum map is a map J : P → → g ∗ ∗ , and the reduced spaces are quotients of the level sets of J by the subgroup of G preserving the level set in question: for μ μ ∈ ∈ g ∗ ∗ one defines P μ μ = J − − 1 ( μ μ ) / G μ μ , and this reduced space is a symplectic manifold if μ μ is a regular value of J.