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**The three-body problem.**
*(English)*
Zbl 0719.70006

Studies in Astronautics, 4. Amsterdam etc.: Elsevier. xvi, 576 p. $ 166.75; Dfl. 325.00 (1990).

The three-body problem is one of the oldest in astronomy. The given monograph is dedicated to this problem. In the last three centuries the three-body problem, i.e. the problem: “What are the free motions of three given spherical bodies moving under the influence of their mutual gravitational attraction?”, has played a major role in the development of mathematics, mechanics, physics and astronomy.

The monograph consists of chapters, and 265 references are given. The first chapters recall the formulations of the three body problem and the main classical results, questions and conjectures. For all this the author easily has shown why the observers of the planets and Moon of antiquity considered astronomy as the mother of science.

The central chapters describe the theory of perturbations and the quantitative analysis of the three body problem recently done. In these chapters we find enough brimful reflection about the modern state of theoretical thoughts and of the power of modern computers. The recent progress in the qualitative analysis on questions such as stability, escapes, singularities, regularizations, final evolutions, periodic motions, asymptotic motions, oscillatory motions, quasicollision motions, Arnold tori, etc. are presented in the final chapters. The remaining open main conjectures and at the same time very important problems are discussed for further investigations in the last chapter.

On the page 563 the author remarks: “It is of course impossible to give even a small idea of the variety and the depth of all these works even in the only domain of the three body problem and even only in U.S. and western Europe.” This is especially true for chapters 8 and 9 of the monograph. In these chapters the works of A. Cook [The motion of the Moon (1988; Zbl 0654.70003)], V. G. Demin, E. P. Aksenov and A. P. Markeev are mentioned. These works were dedicated to the Lagrangian and Eulerian solutions of the three body problem and the motion of the moon.

I am thinking that yet for long time many young people, like the great toiler Victor Szebehely 45 year ago, will be hearing; “Young man, go and solve the problem of three bodies”. And they all will say: “thank you, Christian Marchal, for your book: The three-body problem”, like I am saying today.

The monograph consists of chapters, and 265 references are given. The first chapters recall the formulations of the three body problem and the main classical results, questions and conjectures. For all this the author easily has shown why the observers of the planets and Moon of antiquity considered astronomy as the mother of science.

The central chapters describe the theory of perturbations and the quantitative analysis of the three body problem recently done. In these chapters we find enough brimful reflection about the modern state of theoretical thoughts and of the power of modern computers. The recent progress in the qualitative analysis on questions such as stability, escapes, singularities, regularizations, final evolutions, periodic motions, asymptotic motions, oscillatory motions, quasicollision motions, Arnold tori, etc. are presented in the final chapters. The remaining open main conjectures and at the same time very important problems are discussed for further investigations in the last chapter.

On the page 563 the author remarks: “It is of course impossible to give even a small idea of the variety and the depth of all these works even in the only domain of the three body problem and even only in U.S. and western Europe.” This is especially true for chapters 8 and 9 of the monograph. In these chapters the works of A. Cook [The motion of the Moon (1988; Zbl 0654.70003)], V. G. Demin, E. P. Aksenov and A. P. Markeev are mentioned. These works were dedicated to the Lagrangian and Eulerian solutions of the three body problem and the motion of the moon.

I am thinking that yet for long time many young people, like the great toiler Victor Szebehely 45 year ago, will be hearing; “Young man, go and solve the problem of three bodies”. And they all will say: “thank you, Christian Marchal, for your book: The three-body problem”, like I am saying today.

Reviewer: A.Arazov (Baku)

### MSC:

70F15 | Celestial mechanics |

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |