Memoirs of the American Mathematical Society 2001; 112 pp; softcover Volume: 153 ISBN10: 0821827200 ISBN13: 9780821827208 List Price: US$54 Individual Members: US$32.40 Institutional Members: US$43.20 Order Code: MEMO/153/727
 The perturbation theory of noncommutatively integrable systems is revisited from the point of view of nonAbelian symmetry groups. Using a coordinate system intrinsic to the geometry of the symmetry, we generalize and geometrize wellknown estimates of Nekhoroshev (1977), in a class of systems having almost \(G\)invariant Hamiltonians. These estimates are shown to have a natural interpretation in terms of momentum maps and coadjoint orbits. The geometric framework adopted is described explicitly in examples, including the EulerPoinsot rigid body. Readership Graduate students and research mathematicians interested in topology and algebraic geometry. Table of Contents Part 1. Dynamics  LieTheoretic preliminaries
 Actiongroup coordinates
 On the existence of actiongroup coordinates
 Naive averaging
 An abstract formulation of Nekhoroshev's theorem
 Applying the abstract Nekhoroshev's theorem to actiongroup coordinates
 Nekhoroshevtype estimates for momentum maps
Part 2. Geometry  On Hamiltonian \(G\)spaces with regular momenta
 Actiongroup coordinates as a symplectic crosssection
 Constructing actiongroup coordinates
 The axisymmetric EulerPoinsot rigid body
 Passing from dynamic integrability to geometric integrability
 Concluding remarks
 Appendix A. Proof of the NekhoroshevLochak theorem
 Appendix B. Proof the \({\mathcal W}\) is a slice
 Appendix C. Proof of the extension lemma
 Appendix D. An application of converting dynamic integrability into geometric integrability: The EulerPoinsot rigid body revisited
 Appendix E. Dual pairs, leaf correspondence, and symplectic reduction
 Bibliography
