Introduction
Hamiltonian perturbation theory begins with the construction of a symplec-
tic diffeomorphism between some open subset of the phase space P and an ap-
propriate open subset of a 'model space' M. Presently, action-angle coordinates
(M =
Tn
x
Rn)
and partial action-angle coordinates (M = T
f e
x l
f c
x
M2(n-fc),
0 ^ k n) are the exclusive choices. Both model spaces are canonical in the sense
that, up to a covering, they are Euclidean space equipped with its usual symplectic
structure. Which space is appropriate depends on the nature of the integrability of
the unperturbed system.
Integrability
Classical notions of integrability, cast in terms of integrals of motion and their
Poisson brackets (see, e.g., Arnold et al. (1988)), can be reformulated in geometric
language (Dazord and Delzant, 1987): A Hamiltonian Ho, defined on some sym-
plectic manifold (P, CJ), is integrable if Ho is constant on the leaves of a coisotropic
and symplectically complete foliation J7 on P. A foliation T is coisotropic if its
tangent distribution D contains its symplectic orthogonal Du. It is symplectically
complete if Dw is integrable as a distribution. In that case, the leaves of the corre-
sponding foliation T^ are invariant under the flow of the Hamiltonian vector field
XH0-
To the above 'differential' notion of integrability one adds a global condition.6
The simplest of these, compactness of the leaves of Tu, implies these leaves are
actually fc-dimensional tori and that T^ is an 'angular fiber ing' in the sense of
Nekhoroshev (1972). In particular, in the case of commutative integrability (k
n = (dimP)/2) a neighborhood of each /c-torus admits action-angle coordinates,
while in the case of non-commutative integrability (k n) a neighborhood of each
such torus admits partial action-angle coordinates. In either case, the Hamiltonian
Ho is represented by a function depending only on the k actions 'conjugate' to the
angles representing the tori. This has the consequence that XH0, restricted to a
given /c-torus is a linear vector field (i.e., is 'covered' by a linear vector field on
3Rfe). Proofs of these assertions and further details are given in Dazord and Delzant
(1987).
Generally, in the non-commutative case, it is not possible to construct a partial
action-angle coordinate chart such that it contains a full neighborhood of a leaf of
T. Unfortunately, as Fasso (1995) has pointed out (see also Benettin and Fasso
(1996)), perturbations to Ho create 'fast' motions (motions of the same order as
the perturbation) along the leaves of T. These motions take trajectories out of a
locally defined partial action-angle coordinate chart in a relatively short time. Fasso
6Locally,
all systems are integrable in the stated sense, away from equilibria: Simply apply
Darboux's theorem, taking as first coordinate the Hamiltonian function.
1
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