Constructing action-group coordinates
The local existence of action-group coordinates has been proven by Dazord and
Delzant (1987, Section 5) for suitable non-commutatively integrable systems. We
briefly survey this and related work at the end of Chap. 3. In Part 2 of these memoirs
we offer an alternative construction that we have applied to several mechanical
systems. In this approach the construction of action-group coordinates is reduced
to the construction of conventional action- angle coordinates in an associated lower
dimensional phase space known as a symplectic cross-section. This approach may be
useful to practitioners of perturbation theory, who are already intimately familiar
with conventional action-angle coordinates. The construction may be regarded as a
particular application of the so-called 'symplectic cross-section theorem' (Guillemin,
Lerman and Sternberg, 1996; Guillemin and Sternberg, 1984).
Action-group coordinates can be constructed globally using the cross-section
method precisely when action-angle coordinates can be constructed globally in the
symplectic cross-section (Theorem 10.2, p. 58). Necessary and sufficient conditions
for the existence of global action-angle coordinates in a Hamiltonian system are
already known (Duistermaat, 1980). In Chap. 11 we apply the symplectic cross-
section technique to the axisymmetric Euler-Poinsot rigid body. Perturbations to
this problem have been studied by Benettin and Fasso (1996) using partial action-
angle coordinate charts.
Unfortunately to construct action-group coordinates in the neighborhood of a
point, one must assume that its image under the momentum map is a regular point
of the co-adjoint action. In addition, the symmetry group must be acting freely.
Nevertheless there do exist nontrivial examples for which action-group coordinates
can be constructed, as the axisymmetric Euler-Poinsot rigid body demonstrates.
Other examples include the problem of geodesies on S2 (and hence the 'regular-
ized' 2D Kepler problem10) and the 1 : 1 resonance. Moreover, it seems likely that
techniques such as those outlined here (both the analytic and geometric) will gen-
eralize to cases where more sophisticated Hamiltonian G-space normal forms are
applicable. We discuss this possibility further in Chap. 13.
See Moser (1970).