This volume contains lectures given at the NATO Advanced Study Institute on Long-Time Predictions in Dynamics conducted in Cortina d'Ampezzo, Italy during August 3-16, 1975. The lectures were presented in groups, according to the original structure of the Institute. Under "Fundamentals" the general concepts were treated by Contopoulos, DeWitt, Reichl, Stiefel, Szebehely, Bartlett, Kirchgraber, Verhults and Sigrist. This was followed by the series of lectures on "Numerical and Statistical Analysis" offered by Aarseth, Baumgarte and Tapley. The third principal subject was "Three and Many-Body Problems" with Garfinkel, Broucke, Hadjidemetriou, Marchal, Nahon, Waldvogel, Lasco, and Markellos as the major speakers. The last group of lectures treated "Dynamics in Astronomy" by Colombo, Message, Ovenden, Vicente, and Douglas. Some of the outstanding lectures were rather didactic in nature or were published elsewhere or could not meet the deadline for publication. The Editors will be delighted to furnish leads to those interested in these lectures. Some of the lectures were presented in form of seminar-contributions. These are published as Summaries at the end of this Volume. The Institute was dedicated to the conceptual, analytical, numerical and applied aspects of the problem of long-time predic tion in dynamics. This fundamental problem emerged in all lectures: linearization, regularization, stabilization, averaging, estimation, periodic orbits, qualitative aspects, secular variations, resonance, invariants, etc. were some of the subjects treated in depth. Some conclusions are offered here with the utmost humility and with the advance acknowledgement of the fact that we all hear what we want to hear."

IX LIST OF PRINCIPAL SPEAKERS XI LIST OF PARTICIPANTS 1. REGULARIZATION E. STIEFEL / A Linear Theory of the Perturbed Two-Body Problem (Regul- ization) 3 J. WALDVOGEL / Collision Singularities in Gravitational Problems 21 D. C. HEGGIE / Regularization Using a Time-Transformation Only 34 J. BAUMGAR TE / Stabilization of the Differential Equations of Keplerian Motion 38 F. NAHON / The Particular Solutions of Levi-Civita 45 O. GODAR T / Example ofIntegration of Strongly Oscillating Systems 53 w. BLACK / The Application of Recurrence Relations to Special Perturbation Methods 61 D. G. BETTIS / Numerical Solution of Ordinary Differential Equations (Abstract) 71 II. THE THREE-BODY PROBLEM V. SZEBEHELY / Recent Advances in the Problem of Three Bodies 75 R. F. ARENSTORF / Periodic Elliptic Motion in the Problem of Three Bodies (Abstract) 107 G. KATSIARIS and c. L. GOUDAS / On a Conjecture by Poincare 109 G. KATSIARIS / The Three-Dimensional Elliptic Problem 118 P. G. KAZANTZIS / Second and Third Order Variations of the Three Dimensional Restricted Problem 135 c. G. ZAGOURAS / Planar Periodic Orbits Using Second and Third Variations 146 E. RABE / Elliptic Restricted Problem: Fourth-Order Stability Analysis of the Triangular Points 156 P. GUILLAUME / A Linear Description of the Second Species Solutions 161 III. THE N-BODY PROBLEM AND STELLAR DYNAMICS G. CONTOPOULOS / Problems of Stellar Dynamics 177 w. T. KYNER / Invariant Manifolds in Celestial Mechanics 192 s. J.

This volume contains lectures given at the NATO Advanced Study Institute on Long-Time Predictions in Dynamics conducted in Cortina d'Ampezzo, Italy during August 3-16, 1975. The lectures were presented in groups, according to the original structure of the Institute. Under "Fundamentals" the general concepts were treated by Contopoulos, DeWitt, Reichl, Stiefel, Szebehely, Bartlett, Kirchgraber, Verhults and Sigrist. This was followed by the series of lectures on "Numerical and Statistical Analysis" offered by Aarseth, Baumgarte and Tapley. The third principal subject was "Three and Many-Body Problems" with Garfinkel, Broucke, Hadjidemetriou, Marchal, Nahon, Waldvogel, Lasco, and Markellos as the major speakers. The last group of lectures treated "Dynamics in Astronomy" by Colombo, Message, Ovenden, Vicente, and Douglas. Some of the outstanding lectures were rather didactic in nature or were published elsewhere or could not meet the deadline for publication. The Editors will be delighted to furnish leads to those interested in these lectures. Some of the lectures were presented in form of seminar-contributions. These are published as Summaries at the end of this Volume. The Institute was dedicated to the conceptual, analytical, numerical and applied aspects of the problem of long-time predic tion in dynamics. This fundamental problem emerged in all lectures: linearization, regularization, stabilization, averaging, estimation, periodic orbits, qualitative aspects, secular variations, resonance, invariants, etc. were some of the subjects treated in depth. Some conclusions are offered here with the utmost humility and with the advance acknowledgement of the fact that we all hear what we want to hear."

IX LIST OF PRINCIPAL SPEAKERS XI LIST OF PARTICIPANTS 1. REGULARIZATION E. STIEFEL / A Linear Theory of the Perturbed Two-Body Problem (Regul- ization) 3 J. WALDVOGEL / Collision Singularities in Gravitational Problems 21 D. C. HEGGIE / Regularization Using a Time-Transformation Only 34 J. BAUMGAR TE / Stabilization of the Differential Equations of Keplerian Motion 38 F. NAHON / The Particular Solutions of Levi-Civita 45 O. GODAR T / Example ofIntegration of Strongly Oscillating Systems 53 w. BLACK / The Application of Recurrence Relations to Special Perturbation Methods 61 D. G. BETTIS / Numerical Solution of Ordinary Differential Equations (Abstract) 71 II. THE THREE-BODY PROBLEM V. SZEBEHELY / Recent Advances in the Problem of Three Bodies 75 R. F. ARENSTORF / Periodic Elliptic Motion in the Problem of Three Bodies (Abstract) 107 G. KATSIARIS and c. L. GOUDAS / On a Conjecture by Poincare 109 G. KATSIARIS / The Three-Dimensional Elliptic Problem 118 P. G. KAZANTZIS / Second and Third Order Variations of the Three Dimensional Restricted Problem 135 c. G. ZAGOURAS / Planar Periodic Orbits Using Second and Third Variations 146 E. RABE / Elliptic Restricted Problem: Fourth-Order Stability Analysis of the Triangular Points 156 P. GUILLAUME / A Linear Description of the Second Species Solutions 161 III. THE N-BODY PROBLEM AND STELLAR DYNAMICS G. CONTOPOULOS / Problems of Stellar Dynamics 177 w. T. KYNER / Invariant Manifolds in Celestial Mechanics 192 s. J.