CHAPTER 0

Introduction and Some Salient

Features of the Model Hamiltonian

The search for and study of invariant manifolds is a long established and by now

traditional part of dynamical system theory. In the more restricted but particularly

difficult setting of canonical perturbation theory, the relevance of this approach was

very much enhanced by the discovery of the connection of the splitting of invariant

manifolds with a possible form of global instability in the near-integrable multidi-

mensional Hamiltonian systems. Since this connection was established by V.Arnold

in his famous 1964 note ([Al]) progress has been rather slow and sometimes con-

fused in this difficult area. We hope the reader may get some feeling of the state of

affairs by following the present text and the reading suggestions we have included

all along, particularly of course about the splitting problem, which forms only a

small and perhaps not even unavoidable part of the more general and geometric

question of the global instability. We are not going to try to give an overview of all

the trends and motivations in this introduction. We notice however that the end of

this introduction can offer some clues on the splitting problem and that the closing

sections of Chapters 2 and 3 (§2.6 and §3.7) are partly of expository and indeed

historical nature and can hopefully be used as active surveys of some parts of the

subject.

The present article started in part from the desire to clarify certain basic issues

connected with the splitting of invariant manifolds in the higher dimensional case,

essentially by injecting more geometry into problems which are partly analytic in

essence but do benefit from the introduction of some symplectic geometry, as we

hope will be apparent below. All the more because the problem of global instability

("Arnold diffusion") does indeed require a rather global geometric viewpoint, so

that it seems only natural to start with geometrizing the local situation as much as

possible. Moreover many salient features of this local situation are already apparent

in a deceptively simple-looking problem embodied in the model Hamiltonian (*)

described towards the end of this introduction, which generalizes Arnold's original

example and may serve as a guide into more general situations.

As the reader may have guessed, the present paper is not really meant to be

introductory, and we do occasionally assume some familiarity with the subject on

her or his part. Again the necessary information can easily be gathered from the

reading of recent—or less recent—papers, to which we will point along the way.

Each chapter is provided with its own more specific introduction, which is why this

general introduction has been kept to a minimum. All along we have tried to stress

and sometimes recall some basic and important ideas, occasionally indulging in

some story telling in order to present these ideas in a perhaps more vivid way. This

is in particular at the expense of "effectivity". We have made almost no effort to

compute or estimate most of the constants which appear, although careful reading

will reveal that this is usually painfully but easily doable. We have also at times

been sketchy in the exposition of essentially routine proofs.

We now briefly sketch the content of the paper and then detail some features

of the model problem. Our primary object of study is the splitting of the invariant

Received by the editor July 1, 2000.

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