The Three-Body Problem

1. Summaries. 2. History. 3. The Law of Universal Attraction. 4. Exact Formulations of the Three-body Problem. The classical formulation. The Lagrangian formulation. The Jacobi formulation. The Hamilton and Delaunay formulation. 5. The Invariants in the Three-body Problem. The ten classical integrals and the Lagrange-Jacobi identity. The unsuccessful researches of new integrals. The scale transformation, the variational three-body problem and the eleventh ``local integral''. The integral invariants. 6. Existence and Uniqueness of Solutions. Binary and Triple Collisions. Regularizations of Singularities. 7. Final Simplifications, the Elimination of Nodes, the Elimination of Time. 8. Simple Solutions of the Three-body Problem. The Lagrangian and Eulerian solutions. The central configurations. Stability of Eulerian and Lagrangian motions. The Eulerian and Lagrangian motions in nature and astronautics. Other exact solutions of the three-body problem. Other simple solutions of the three-body problem. 9. The Restricted Three-body Problem. The circular restricted three-body problem. The Hill problem. The elliptic, parabolic and hyperbolic restricted three-body problems. The Copenhagen problem and the computations of Michel Hénon. 10. The General Three-body Problem. Quantitative Analysis. The analytical methods. An example of the Von Zeipel method. Integration of the three-body problem to the first order. Integration of the three-body problem to the second order. The numerical methods. Periodic orbits and numerical methods. Periodic orbits and symmetry properties. The vicinity and the stability of periodic orbits. The series of some simple solutions of the three-body problem. Examples of numerical integrations. 11. The General Three-body Problem. Qualitative Analysis and Qualitative Methods. The prototype of qualitative methods. The trivial transformations and the corresponding symmetries among n-body orbits. Other early qualitative researches. Periodic orbits. The method of Poincaré. Unsymmetrical periodic orbits. The Brown conjecture. The Hill stability and its generalization. Final evolutions and tests of escape. The n-body motions and complete collapses. An extension of the Sundman three-body result. Original and final evolutions. On the Kolmogorov-Arnold-Moser theorem. The Arnold diffusion conjecture. The temporary chaotic motions. The temporary capture. An application of qualitative methods. The controversy between Mrs. Kazimirchak-Polonskaya and Mr. R. Dvorak. The Lagrangian and the qualitative methods. 12. Main Conjectures and Further Investigations. Conclusions. Appendices. References. Bibliography. Subject index. Author index.