## The three-body problemRecent 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. |

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### Contents

Summaries in eight languages | 1 |

History | 12 |

The invariants in the threebody problem | 21 |

Copyright | |

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### Common terms and phrases

all-order stability analysis analytic angular momentum Arnold diffusion asymptotic binary bodies Celestial Mechanics center of mass circular restricted three-body conjecture constant corresponding defined Delaunay eccentricity eigenvalues elliptic energy integral equal masses equations of motion Eulerian motions exponential families of periodic final evolution Floquet multipliers given Halo orbits Hamiltonian system Hence Hill-type stability implies infinitesimal masses infinity instability instance integral h integrals of motion Jacobi integral Keplerian Lagrangian motions Lagrangian points leads Let us consider limit lower bound mass ratios mean quadratic distance monodromy matrix mutual distance n-body problem orbit of interest oscillatory parameters pericenter periodic orbits periodic solution perturbations planar plane Poincare possible power series quasi-integrals resonance restricted three-body problem satisfied semi-major axis set of axes solution of interest space-time symmetry tests of escape theorem three masses three-body motions transformation triple collision two-body values vectors velocity vicinity