Small Denominators and Problems of Stability of Motion in Classical and Celestial Mechanics
Abstract
CONTENTSIntroduction § 1. Results § 2. Preliminary results from mechanics § 3. Preliminary results from mathematics § 4. The simplest problem of stability § 5. Contents of the paperChapter 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 systemsChapter 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 caseChapter 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 \Bar{\Bar F}_1 § 5. The many-body problemChapter 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 theoremChapter V. Technical lemmas § 1. Domains of type D § 2. Arithmetic lemmas § 3. Analytic lemmas § 4. Geometric lemmas § 5. Convergence lemmas § 6. NotationChapter VI. Appendix § 1. Integrable systems § 2. Unsolved problems § 3. Neighbourhood of an invariant manifold §4. Intermixing § 5. Smoothing techniquesReferences
- Publication:
-
Russian Mathematical Surveys
- Pub Date:
- December 1963
- DOI:
- 10.1070/RM1963v018n06ABEH001143
- Bibcode:
- 1963RuMaS..18R...2A