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