The perturbation theory of non-commutatively integrable systems is revisited
from the point of view of non-Abelian symmetry groups. Using a coordinate system
intrinsic to the geometry of the symmetry, we generalize and geometrize well-known
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 co-adjoint orbits. The geometric framework adopted is
described explicitly in examples, including the Euler-Poinsot rigid body.
Received by the editor May 26, 1998 and in revised form April 14, 2000.
The author thanks Jerry Marsden for considerable help and support during this project.
Helpful feedback has been supplied by Francesco Fasso, Tudor Ratiu and Pierre Lochak.
Key words and phrases. Hamiltonian perturbation theory, non-Abelian symmetry, action-
angle coordinates, non-canonical coordinates, rigid body, Nekhoroshev estimate, momentum map,
moment map, co-adjoint orbit, symplectic cross-section, action-group coordinates, Hamiltonian
2000 Mathematics Subject Classification. Primary 70H09 70H08 37J15 ; Secondary 37J40
70H33 70H06.
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