Hamiltonian group actions and integrable systems

The purpose of this paper is to show how some ideas from the theory of Hamiltonian systems with symmetry can be applied to dynamical systems like the nonperiodic Toda Lattice of Moser to yield striking information about their trajectories.Our presentation of these ideas is as self-contained as possible. We try to keep abstract machinery in its proper place, and emphasize the concrete computations which are implicit in some parts of modern symplectic geometry. Most of the results we shall derive are known in one form or another. The only novelties, perhaps, are some proofs and examples.