THE THREE-BODY PROBLEM

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1 STUDIES IN ASTRONAUTICS 4 THE THREE-BODY PROBLEM CHRISTIAN MARCH AL Office National d'etudes et de RecherchesAerospatiales, Chätillon, France Amsterdam - Oxford - New York -Tokyo 1990

2 X CONTENTS Foreword Dedication Acknowledgments Short table of Contents contents 1.Summaries (English, French, Russian, German, Spanish, Japanese, Chinese, Arabic) 2.History 3.The law of universal attraction 4.Exact formulations of the three-body problem 4.1.The classical formulation 4.2.The Lagrangian formulation 4.3.The Jacobi formulation 4.4.The Hamilton and Delaunay formulation 5.The invariants in the three-body problem 5.1.The ten classical integrals and the Lagrange-Jacobi identity The integral of the center of mass The integral of angular momentum The integral of the energy The Lagrange-Jacobi identity 5.2.The unsuccessful researches of new integrals 5.3.The scale transformation, the variational threebody problem and the eleventh "local integral" 5.4.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 8.1.The Lagrangian and Eulerian solutions. The central configurations

3 XI 8.2.Stability of Eulerian and Lagrangian motions First-order analysis Complete analysis of stability The Eulerian and Lagrangian motions in nature and in astronautics Other exact solutions of the three-body problem The isoceles solutions The z-axis Hill solutions Other simple solutions of the three-body problem 54 9.The restricted three-body problem The circular restricted three-body problem The Hill problem The Brown series The lunar motion within 1000 km The elliptic, parabolic and hyperbolic restricted three-body problems The Copenhagen problem and the computations of Michel Henon 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 Principle of the method of Von Zeipel Application of the method of Von Zeipel to the three-body problem First-order integration of the three-body problem A concrete picture of the wide perturbations of the three-body problem General considerations on the first-order integration Integration of the three-body problem to the second order The numerical methods A three-body motion of the exchange type An oscillatory motion of the second kind Studies of gravitational scattering Periodic orbits and numerical methods Computation of periodic orbits. The method of analytic continuation. The utmost reduction of the three-body problem and the elimination of

4 Xll trivial side-effects The method of analytic continuation for three given masses The method of analytic continuation and the modification of masses Per iodic orbits and symmetry properties The four types of space-time symmetries Families of symmetric periodic orbits The vicinity and the stability of periodic orbits Definition and generalities The evolution of ignorable parameters. The orbital stability. The "in plane" stability The first-order analysis Simple cases of the first-order analysis Rectilinear periodic orbits Plane periodic orbits Symmetric periodic orbits Circular restricted case and Hill case First-order stability, the general discussion On the evolution of first-order stability along the families of periodic orbits Elements of the all-order stability analysis. The near-resonance theorem Analytic autonomous differential systems. The vicinity of a point of equilibrium Analytic differential systems. The vicinity of a periodic solution Mot ions in the central subset. Motions in the critical case. The critical Hamiltonian case Critical Hamiltonian case. The N th -order study. The quasi-integrals. Generalization of "Birkhoff differential rotations" The six main types of stability and instability A lower bound of m for a "power-m instability" Two conjectures on the stability or instability of periodic solutions of analytic Hamiltonian systems 217

5 On the cases with multiple Floquet multipliers or multiple eigenvalues Example. The all-order stability of Lagrangian motions The first order study The second simplification The quasi-integrals I N Ex tension to the circular Lagrangian motions of the general three-body problem The second-order study The third-order study The series of some simple solutions of the threebody problem The pseudo-circular orbits A family of periodic orbits with the largest number of symmetries The Halo orbits about the collinear Lagrangian points Examples of numerical integrations Researches by continuity The retrograde pseudo-circular orbits of the three-body problem with three equal masses A numerical experiment. The Pythagorean problem The method of numerical exploration. Encounters of satellites 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 The space-time symmetries The space symmetries The remaining symmetries Multi-symmetries Other early qualitative researches The Eulerian and Lagrangian solutions. The central configurations The research of new integrals of motion Periodic orbits. The method of Poincare 317

6 The three first species of Poincare periodic orbits The Poincare conjecture Unsymmetrical periodic orbits. The Brown conjecture The Hill stability and its generalization The "generalized semi-major axis", the "generalized semi-latus rectum", the "mean quadratic distance", the "mean harmonic distance" and the Sundman function The classical relations and the new notation Hill-type stability in the general threebody problem Scale effects Hill-type stability for systems with positive or zero energy integral Final evolutions and tests of escape The new notations and the n-body problem The classical results and the new notations Improvements - (Three and n-body motions) Limitations on the configuration, the scale, the orientation On the evolution of the semi-moment of inertia I and the mean quadratic distance p On the evolution of the potential U and the mean harmonic distance "V A psychological improvement, the use of К and Л instead of p and v The principle of the tests of escape Example of the construction of a test of escape for the n-body problem Simplification of the problem Research of long-term valid results Improvement of the efficiency of the test. Extension to the general n-body problem Final evolution : the singularities The two types of singularity of the n-body problem Impossibility of the "infinite expansion

7 XV in a bounded interval of time" for threebody motions Analysis of a collision Collisions and central configurations On the regularization of singularities Final evolutions. The Chazy classification of three-body motions Relations among the lengths X. The limits of the vectors r- /t 401 о The hyperbolic final evolution The hyperbolic-parabolic and the hyperbolic-elliptic final evolutions The tri-parabolic final evolution The parabolic-elliptic final evolution The bounded evolution, the two oscillatory evolutions and the collisions of stars Sitnikov motions and oscillatory evolutions of the first kind General table of final evolutions Progress in the tests of escape Classification of tests The ergodic theorem. The difficulty of a test of bounded motions A test of escape valid even for very small mutual distances An application of the very efficient test. Analysis in the (p, p') half-plane A survey of recent progress in tests of escape. Analysis of triple close approaches n-body motions and complete collapses. An extension of the Sundman three-body result Original and final evolutions General three-body systems of positive energy and non-zero angular momentum General three-body systems of positive energy and zero angular momentum General three-body systems of zero energy and non-zero angular momentum General three-body systems of zero energy and

8 zero angular momentum General three-body systems of negative energy and non-zero angular momentum Remaining cases. Restricted cases 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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