Outline of Part 1
Part 1 of this memoir focuses on dynamics. After dispensing in Chap. 1 with
some Lie theoretic preliminaries, we begin in Chap. 2 with a resume of the action-
group coordinate framework. In Chap. 3 we state conditions ensuring the existence
of action-group coordinates in a given system (Theorem 3.10, p. 19). We include a
description of geodesic motions on S*2, where the existence of action-group coordi-
nates is fairly transparent.
Chap. 4 describes the geometry of the unperturbed dynamics of a system in
action-group coordinates, as well as the dynamics obtained after naively applying
'multi-phase averaging' to the perturbation. This will explain, from the symmetry
point of view, the origin of the 'fast' motions tangent to the co-adjoint orbits,
and motivate the Nekhoroshev estimates for the transverse motions (Corollary 7.1,
p. 43), which we deduce in Sections 5-7. These estimates follow from Theorem
5.8 (p. 30), an abstract generalization of Nekhoroshev's theorem that we prove in
Appendix A using the method of Lochak (1992).
Some limitations, as well as opportunities for further investigation, are dis-
cussed in Chap. 13.
Outline of Part 2
Part 2 describes geometric constructions underlying the analyses given in Part
1. Chap. 8 studies a special class of Hamiltonian G-spaces, namely those possessing
points of 'regular co-adjoint orbit type.' It is shown (Theorem 8.14, p. 50) that all
geometric information describing such a space is encoded in a certain Hamiltonian
T-space of lower dimension (its symplectic cross-section).
In Chap. 9 we show how the action-group model space can be realized as a
symplectic submanifold of T*G. We derive expressions for the symplectic structure
and for Hamiltonian vector fields stated Chap. 2.
In Chap. 10 we use the results of Chap. 8 to reduce the problem of constructing
action-group coordinates in an integrable system, to the construction of action- angle
coordinates in its symplectic cross-section (Theorem 10.2, p. 58). We tackle the
problem of global existence by relating work of Duistermaat (1980). We also prove
'semi-global' results, which are more convenient to apply in concrete examples. This
includes a proof of Theorem 3.10 stated in Chap. 3 (see Corollary 10.12, p. 63). An
application to the axisymmetric Euler-Poinsot rigid body is made in Chap. 11.
The constructions of Chap. 10 apply to systems that have zero dimensional
Marsden-Weinstein reduced spaces, with respect to some known symmetry group
G (we call this geometric integrability). In Chap. 12 we show how one can sometimes
enlarge G in a system with two dimensional reduced spaces (dynamic integrability)
to G' = G x S1, in such a way as to render the G'-reduced spaces zero dimensional.
An application to rigid bodies is given in Appendix D.
In Appendix E we include a detailed proof of Weinstein's Symplectic Leaf
Correspondence Theorem, which we use in Chap. 12.
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