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
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 5.1.1.The integral of the center of mass 5.1.2.The integral of angular momentum 5.1.3.The integral of the energy 5.1.4.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
XI 8.2.Stability of Eulerian and Lagrangian motions 44 8.2.1.First-order analysis 46 8.2.2.Complete analysis of stability 50 8.3.The Eulerian and Lagrangian motions in nature and in astronautics 51 8.4.Other exact solutions of the three-body problem 53 8.4.1.The isoceles solutions 53 8.4.2.The z-axis Hill solutions 54 8.5.Other simple solutions of the three-body problem 54 9.The restricted three-body problem 58 9.1.The circular restricted three-body problem 59 9.2.The Hill problem 63 9.2.1.The Brown series 64 9.2.2.The lunar motion within 1000 km 66 9.3.The elliptic, parabolic and hyperbolic restricted three-body problems 68 9.4.The Copenhagen problem and the computations of Michel Henon 71 10.The general three-body problem. Quantitative analysis 79 10.1.The analytical methods 79 10.2.An example of the Von Zeipel method. Integration of the three-body problem to the first order 81 10.2.1.Principle of the method of Von Zeipel 83 10.2.2.Application of the method of Von Zeipel to the three-body problem 84 10.2.3.First-order integration of the three-body problem 87 10.2.4.A concrete picture of the wide perturbations of the three-body problem 93 10.2.5.General considerations on the first-order integration 96 10.3.Integration of the three-body problem to the second order 98 10.4.The numerical methods 99 10.4.1.A three-body motion of the exchange type 101 10.4.2.An oscillatory motion of the second kind 103 10.4.3.Studies of gravitational scattering 106 10.5.Periodic orbits and numerical methods 118 10.5.1.Computation of periodic orbits. The method of analytic continuation. The utmost reduction of the three-body problem and the elimination of
Xll trivial side-effects 123 10.5.2.The method of analytic continuation for three given masses 127 10.5.3.The method of analytic continuation and the modification of masses 134 10.6. Per iodic orbits and symmetry properties 136 10.6.1.The four types of space-time symmetries 136 10.6.2.Families of symmetric periodic orbits 141 10.7.The vicinity and the stability of periodic orbits 146 10.7.1.Definition and generalities 146 10.7.2.The evolution of ignorable parameters. The orbital stability. The "in plane" stability 149 10.7.3.The first-order analysis 150 10.7.4.Simple cases of the first-order analysis 158 10.7.4.1.Rectilinear periodic orbits 158 10.7.4.2.Plane periodic orbits 161 10.7.4.3.Symmetric periodic orbits 164 10.7.4.4.Circular restricted case and Hill case 166 10.7.5.First-order stability, the general discussion 169 10.7.6.On the evolution of first-order stability along the families of periodic orbits 171 10.7.7.Elements of the all-order stability analysis. The near-resonance theorem 174 10.7.7.1.Analytic autonomous differential systems. The vicinity of a point of equilibrium 174 10.7. 7. 2. Analytic differential systems. The vicinity of a periodic solution 184 10.7.7.3.Mot ions in the central subset. Motions in the critical case. The critical Hamiltonian case 194 10.7.7.4.Critical Hamiltonian case. The N th -order study. The quasi-integrals. Generalization of "Birkhoff differential rotations" 206 10.7.7.5.The six main types of stability and instability 213 10.7.7.6.A lower bound of m for a "power-m instability" 216 10.7.8.Two conjectures on the stability or instability of periodic solutions of analytic Hamiltonian systems 217
10.7.9.On the cases with multiple Floquet multipliers or multiple eigenvalues 221 10.7.10.Example. The all-order stability of Lagrangian motions 222 10.7.10.1.The first order study 224 10.7.10.2.The second simplification 225 10.7.10.3.The quasi-integrals I N 227 10.7.10.4.Ex tension to the circular Lagrangian motions of the general three-body problem 231 10.7.10.5.The second-order study 237 10.7.10.6.The third-order study 245 10.8.The series of some simple solutions of the threebody problem 248 10.8.1.The pseudo-circular orbits 249 10.8.2.A family of periodic orbits with the largest number of symmetries 251 10.8.3.The Halo orbits about the collinear Lagrangian points 257 10.9.Examples of numerical integrations 277 10.9.1.Researches by continuity The retrograde pseudo-circular orbits of the three-body problem with three equal masses 277 10.9.2.A numerical experiment. The Pythagorean problem 284 10.9.3.The method of numerical exploration. Encounters of satellites 291 11.The general three-body problem. Qualitative analysis and qualitative methods 301 11.1.The prototype of qualitative methods 301 11.2.The trivial transformations and the corresponding symmetries among n-body orbits 302 11.2.1.The space-time symmetries 305 11.2.2.The space symmetries 309 11.2.3.The remaining symmetries 310 11. 2.4.Multi-symmetries 311 11.3.Other early qualitative researches 311 11.3.1.The Eulerian and Lagrangian solutions. The central configurations 311 11.3.2.The research of new integrals of motion 316 11.4.Periodic orbits. The method of Poincare 317
11.4.1.The three first species of Poincare periodic orbits 317 11.4.2.The Poincare conjecture 321 5.Unsymmetrical periodic orbits. The Brown conjecture 322 6.The Hill stability and its generalization 323 11.6.1.The "generalized semi-major axis", the "generalized semi-latus rectum", the "mean quadratic distance", the "mean harmonic distance" and the Sundman function 326 11.6.2.The classical relations and the new notation 327 11. 6.3.Hill-type stability in the general threebody problem 329 11.6.4.Scale effects 338 11. 6. 5.Hill-type stability for systems with positive or zero energy integral 339 7.Final evolutions and tests of escape 341 11.7.1.The new notations and the n-body problem 341 11.7.2.The classical results and the new notations 344 11.7.3.Improvements - (Three and n-body motions) 347 11.7.3. 1. Limitations on the configuration, the scale, the orientation 347 11.7.3.2.On the evolution of the semi-moment of inertia I and the mean quadratic distance p 349 11.7.3.3.On the evolution of the potential U and the mean harmonic distance "V 358 11.7.3.4.A psychological improvement, the use of К and Л instead of p and v 361 11.7.4.The principle of the tests of escape 369 11.7.5.Example of the construction of a test of escape for the n-body problem 371 11.7.5.1.Simplification of the problem 372 11.7.5.2.Research of long-term valid results 373 11.7.5.3.Improvement of the efficiency of the test. Extension to the general n-body problem 377 11.7.6.Final evolution : the singularities 383 11.7.6.1.The two types of singularity of the n-body problem 384 11.7.6.2.Impossibility of the "infinite expansion
XV in a bounded interval of time" for threebody motions 385 11. 7.6.3. Analysis of a collision 387 11.7.6.4.Collisions and central configurations 390 11.7.6.5.On the regularization of singularities 396 11.7.7.Final evolutions. The Chazy classification of three-body motions 398 11. 7. 7. 1. Relations among the lengths X. The limits of the vectors r- /t 401 о 11.7.7.2.The hyperbolic final evolution 403 11.7.7.3.The hyperbolic-parabolic and the hyperbolic-elliptic final evolutions 404 11.7.7.4.The tri-parabolic final evolution 406 11.7.7.5.The parabolic-elliptic final evolution 408 11.7.7.6.The bounded evolution, the two oscillatory evolutions and the collisions of stars 410 11.7.8.Sitnikov motions and oscillatory evolutions of the first kind 419 11.7.9. General table of final evolutions 424 11.7. 10.Progress in the tests of escape 428 11. 7. 10.1.Classification of tests 429 11. 7. 10. 2. The ergodic theorem. The difficulty of a test of bounded motions 432 11.7.10.3.A test of escape valid even for very small mutual distances 436 11.7.10.4.An application of the very efficient test. Analysis in the (p, p') half-plane 454 11.7.10.5.A survey of recent progress in tests of escape. Analysis of triple close approaches 483 11.8.n-body motions and complete collapses. An extension of the Sundman three-body result 489 11.9. Original and final evolutions 493 11. 9. 1. General three-body systems of positive energy and non-zero angular momentum 494 11.9.2.General three-body systems of positive energy and zero angular momentum 495 11.9.3.General three-body systems of zero energy and non-zero angular momentum 496 11.9.4.General three-body systems of zero energy and
zero angular momentum 11.9.5.General three-body systems of negative energy and non-zero angular momentum 11. 9.6. Remaining cases. Restricted cases 11.10.On the Kolmogorov-Arnold-Moser theorem 11.11.The Arnold diffusion conjecture. The temporary chaotic motions. The temporary capture 11.12.An application of qualitative methods. The controversy between Mrs Kazimirchak-Polonskaya and Mr R. Dvorak 11.13.The Lagrangian and the qualitative methods 12.Main conjectures and further investigations Conclusions Appendices References Bibliography Subject index Author index 497 499 506 507 509 513 517 519 523 527 547 563 566 570