overcomes this problem by showing how to make intrinsic sense of normal forms for
the perturbed Hamiltonian. Such normal forms are used to deduce Nekhoroshev
estimates on other components of the motion, although to compute and express
such estimates a particular atlas of partial action-angle coordinate charts must be
fixed. The extent to which these estimates depend on the choice is an open problem
(see Fasso (1995, Appendix C)).
Another approach to non-commutative integrability, due to Mishchenko and
Fomenko (1978) (see also Arnold et al. (1988)), is to convert non-commutative in-
tegrability into commutative integrability. This is accomplished, at least locally,
by 'gluing' together the /c-tori to form n-tori; out of these n-tori one builds con-
ventional action-angle coordinates. There is no escaping the difficulties mentioned
above, however, because the unperturbed Hamiltonian will still depend on only
k n actions, leading to 'fast' motions in the remaining variables which take
trajectories out of locally defined coordinate charts as before. Moreover, this de-
generacy of the Hamiltonian appears mysteriously because its source, namely the
non-commutative geometry of the original problem, is concealed by the 'commut-
ative' coordinates employed.
Integrability throug h s y m m e t r y
Non-commutative integrability arises frequently in applications and is often
manifest in the form of a non-Abelian symmetry group G. In that case the leaves
of T correspond to orbits of G in the phase space7. Generally the geometry of
the symmetry is obscured in partial action-angle coordinate charts since, as we
remarked above, such charts need not contain full neighborhoods of leaves of J78.
An alternative that we explore here is to substitute for a canonical model space
a Hamiltonian G-space normal form. One can view these normal forms as 'non-
canonical coordinates' built directly out of the non-Abelian group action. The
simplest of these normal forms is a generalization of action-angle coordinates that
we shall refer to as action-group coordinates ( M = G x tj$; see Chap. 1). The 'group'
in action-group coordinates can be any compact connected Lie group G; when G =
T n , one recovers conventional action-angle coordinates. In action-group coordinates
the /c-tori discussed above are represented by cosets in G of some maximal torus
T c G .
Action-group coordinates appear in Dazord and Delzant (1987, Section 5).
Forerunners of these coordinates have been obtained by Marsden (1981), Got ay
(1982), Marie (1983a) and Guillemin and Sternberg (1984, §41). We shall see
that in the perturbation analysis of a non-Abelian symmetry, a single action-group
coordinate chart suffices, and applies under conditions that are readily verified.
Generalized Nekhoroshe v estimate s
Our main objective is to demonstrate that a Hamiltonian G-space normal form
can indeed be used as a geometric framework for perturbation theory. We do
this in Part 1 by generalizing a Nekhoroshev's (1977) theorem to a 'non-canonical'
setting that includes action-group coordinates (Theorem 5.8, p. 30). As a corollary
7 A proof of this fact will be recalled in Remark 3.12.
In fact (see Remark 4.4), if G acts freely and is compact connected and non-Abelian, then it
is never possible for a partial action-angle coordinate chart to contain a neighborhood of a G-orbit
(i.e., leaf of T).