Table of contents

CONTENTS Introduction Chapter I. Formal groups and Dieudonné modules; basic concepts 1. Groups in categories 2. Algebraic and formal groups. Bialgebras 3. The structure of commutative artinian groups 4. The Dieudonné module of a formal group 5. Comments Chapter II. Dieudonné modules; classification up to isogeny 1. Reduction of the problem 2. Modules over the ring A 3. A technical result 4. Classification of formal groups up to isogeny 5. Comments Chapter III. Dieudonné modules; classification up to isomorphism 1. Statement of the problem 2. Auxiliary results 3. The algebraic structure on the module space 4. The structure of isosimple modules; subsidiary reduction 5. The structure of isosimple modules; proof of the first finiteness theorem 6. The second finiteness theorem 7. Cyclic isosimple modules; the component of maximal dimension 8. Classification of two-dimensional modules 9. Comments Chapter IV. Algebroid formal groups and abelian varieties 1. General results 2. The formal structure of abelian varieties; preliminary reduction 3. The formal structure of abelian varieties; the fundamental theorem 4. Weakly algebroid groups 5. Remarks and examples 6. Comments References

CONTENTS Introduction § 1. Results § 2. Preliminary results from mechanics § 3. Preliminary results from mathematics § 4. The simplest problem of stability § 5. Contents of the paper Chapter I. Theory of perturbations § 1. Integrable and non-integrable problems of dynamics § 2. The classical theory of perturbations § 3. Small denominators § 4. Newton's method § 5. Proper degeneracy § 6. Remark 1 § 7. Remark 2 § 8. Application to the problem of proper degeneracy § 9. Limiting degeneracy. Birkhoff's transformation § 10. Stability of positions of equilibrium of Hamiltonian systems Chapter II. Adiabatic invariants § 1. The concept of an adiabatic invariant § 2. Perpetual adiabatic invariance of action with a slow periodic variation of the Hamiltonian § 3. Adiabatic invariants of conservative systems § 4. Magnetic traps § 5. The many-dimensional case Chapter III. The stability of planetary motions § 1. Picture of the motion § 2. Jacobi, Delaunay and Poincaré variables §3. Birkhoff's transformation § 4. Calculation of the asymptotic behaviour of the coefficients in the expansion of § 5. The many-body problem Chapter IV. The fundamental theorem § 1. Fundamental theorem § 2. Inductive theorem § 3. Inductive lemma § 4. Fundamental lemma § 5. Lemma on averaging over rapid variables § 6. Proof of the fundamental lemma § 7. Proof of the inductive lemma § 8. Proof of the inductive theorem § 9. Lemma on the non-degeneracy of diffeomorphisms § 10. Averaging over rapid variables § 11. Polar coordinates § 12. The applicability of the inductive theorem § 13. Passage to the limit § 14. Proof of the fundamental theorem Chapter V. Technical lemmas § 1. Domains of type D § 2. Arithmetic lemmas § 3. Analytic lemmas § 4. Geometric lemmas § 5. Convergence lemmas § 6. Notation Chapter VI. Appendix § 1. Integrable systems § 2. Unsolved problems § 3. Neighbourhood of an invariant manifold §4. Intermixing § 5. Smoothing techniques References