On the Three-body Problem and the Equations of Dynamics

Abstract

Here is an accurate and readable translation of a seminal article by Henri Poincaré that is a classic in the study of dynamical systems popularly called chaos theory. In an effort to understand the stability of orbits in the solar system, Poincaré applied a Hamiltonian formulation to the equations of planetary motion and studied these differential equations in the limited case of three bodies to arrive at properties of the equations? solutions, such as orbital resonances and horseshoe orbits. Poincaré wrote for professional mathematicians and astronomers interested in celestial mechanics and differential equations. Contemporary historians of math or science and researchers in dynamical systems and planetary motion with an interest in the origin or history of their field will find his work fascinating. . Translator's Preface; The Monograph; Errors and Typos; The Translation; References and Index; Notation; Section 4 of Chapter 2, Theorem III; Notable Concepts; Acknowledgements; Bibliography; Author's Preface; Contents; Review; 1 General Properties of the Differential Equations; 1 Notations and Definitions; 2 Calculation of Limits; 3 Application of the Calculation of Limits to Partial Differential Equations; 4 Integration of Linear Equations with Periodic Coefficients; 2 Theory of Integral Invariants; 1 Various Properties of the Equations of Dynamics; 2 Definitions of Integral Invariants 3 Transformation of Integral Invariants4 Using Integral Invariants; 3 Theory of Periodic Solutions; 1 Existence of Periodic Solutions; 2 Characteristic Exponents; 3 Periodic Solutions of the Equations of Dynamics; 4 Calculation of the Characteristic Exponents; 5 Asymptotic Solutions; 6 Asymptotic Solutions of the Equations of Dynamics; Equations of Dynamics and the N-Body Problem; 4 Study of the Case with Only Two Degrees of Freedom; 1 Various Geometric Representations; 5 Study of the Asymptotic Surfaces; 1 Description of the Problem; 2 First Approximation; 3 Second Approximation 4 Third Approximation6 Various Results; 1 Periodic Solutions of the Second Kind; 2 Divergence of Lindstedt's Series; 3 Nonexistence of One-to-One Integrals; 7 Attempts at Generalization; 1 The N-Body Problem; Erratum; Bibliography; Author Index; Subject Index