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**Capture dynamics and chaotic motions in celestial mechanics. With applications to the construction of low energy transfers.**
*(English)*
Zbl 1057.70001

Princeton, NJ: Princeton University Press (ISBN 0-691-09480-2/hbk). xvii, 211 p. (2004).

The main topic is to study the complicated dynamics in the three-body problem, and also to present some new results by the author and collaborators in a unified manner, with special emphasis on capture dynamics. The first chapter includes an introduction to celestial mechanics and two- and three-body problems, and a new proof of existence of an equivalent geodesic flow for the two-body problem. The theory of bounded motion in Hamiltonian systems and the study of the associated symplectic twist map is presented in a short but self-contained way in the second chapter. The main theorems are presented, and the dynamics in the resonant zones is described. The chapter concludes with an application of Aubry-Mather theory to the restricted problem. The capture problem is defined in the third chapter. Different types of capture are also defined, and astrodynamic applications are discussed. This chapter includes several new theorems not previously published. It is shown that two different types of capture defined in a completely different way, i.e. permanent and ballistic capture, are equivalent on a special set whose main feature is the existence of a hyperbolic network. Special emphasis is given to the theoretical approach, although lengthy proofs of some theorems are omitted. In these cases, however, a sketch of the proof is included, and the pertinent references are always supplied. The applications are well illustrated with expository examples from astronomy or astrodynamics. In conclusion, although the book is mainly a textbook on celestial mechanics, it can also be used as an introductory course in Hamiltonian dynamical systems and the three-body problem, while it is at the same time a valuable research tool in the particular field of capture dynamics.

Reviewer: Simos Ichtiaroglou (Thessaloniki)

### MSC:

70-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of particles and systems |

70F15 | Celestial mechanics |

70F07 | Three-body problems |

37N05 | Dynamical systems in classical and celestial mechanics |

70H12 | Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics |