The Three-Body Problem book cover

The Three-Body Problem

Recent research on the theory of perturbations, the analytical approach and the quantitative analysis of the three-body problem have reached a high degree of perfection. The use of electronics has aided developments in quantitative analysis and has helped to disclose the extreme complexity of the set of solutions. This accelerated progress has given new orientation and impetus to the qualitative analysis that is so complementary to the quantitative analysis.

The book begins with the various formulations of the three-body problem, the main classical results and the important questions and conjectures involved in this subject. The main part of the book describes the remarkable progress achieved in qualitative analysis which has shed new light on the three-body problem. It deals with questions such as escapes, captures, periodic orbits, stability, chaotic motions, Arnold diffusion, etc. The most recent tests of escape have yielded very impressive results and border very close on the true limits of escape, showing the domain of bounded motions to be much smaller than was expected. An entirely new picture of the three-body problem is emerging, and the book reports on this recent progress.

The structure of the solutions for the three-body problem lead to a general conjecture governing the picture of solutions for all Hamiltonian problems. The periodic, quasi-periodic and almost-periodic solutions form the basis for the set of solutions and separate the chaotic solutions from the open solutions.

Included in series
Studies in Astronautics

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Published: July 1990

Imprint: Elsevier

ISBN: 978-0-444-87440-5

Reviews

  • This work will be the standard reference work...for many years to come.
    INDA Journal of Nonwovens Research


    ...many young people...will be hearing; "Young man, go and solve the problem of three bodies". And they all will say: "thank you...for your book...like I am saying today...
    Zentralblatt fuer Mathematik
    There is no question; this was, indeed, a most ambitious project!
    Celestial Mechanics and Dynamical Astronomy

Contents

  • 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.

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