Three body problem
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The mathematical theory entered a new era at the end of the 19th century with the works of Poincaré and since the 1950s with the development of computers.Abstract:
The problem is to determine the possible motions of three point masses \\(m_1\\ ,\\) \\(m_2\\ ,\\) and \\(m_3\\ ,\\) which attract each other according to Newton's law of inverse squares. It started with the perturbative studies of Newton himself on the inequalities of the lunar motion[1]. In the 1740s it was constituted as the search for solutions (or at least approximate solutions) of a system of ordinary differential equations by the works of Euler, Clairaut and d'Alembert (with in particular the explanation by Clairaut of the motion of the lunar apogee). Much developed by Lagrange, Laplace and their followers, the mathematical theory entered a new era at the end of the 19th century with the works of Poincaré and since the 1950s with the development of computers. While the two-body problem is integrable and its solutions completely understood (see [2],[AKN],[Al],[BP]), solutions of the three-body problem may be of an arbitrary complexity and are very far from being completely understood.read more
Citations
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The Eccentric Kozai-Lidov Effect and Its Applications
TL;DR: The hierarchical triple-body approximation has useful applications to a variety of systems from planetary and stellar scales to supermassive black holes as mentioned in this paper, where the energy of each orbit is separately conserved, and therefore the two semimajor axes are constants.
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Secular Dynamics in Hierarchical Three-Body Systems
Smadar Naoz,Smadar Naoz,Will M. Farr,Yoram Lithwick,Frederic A. Rasio,Jean Teyssandier,Jean Teyssandier +6 more
TL;DR: In this paper, the authors derive the evolution equations to octupole order in Hamiltonian perturbation theory for hierarchical triple systems, and show new behaviors including the possibility for a system to oscillate from prograde to retrograde orbits.
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The Analytical Foundations of Celestial Mechanics
TL;DR: Wintner as discussed by the authors discusses the need for a calendar from the beginning of civilization, namely the need to make itself felt, namely, for some form of calendar, which gives rise to the primitive study of astronomy, involving some knowledge of the sun and the moon.
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Post-Oligarchic Evolution of Protoplanetary Embryos and the Stability of Planetary Systems
TL;DR: In this paper, the authors investigated the orbit-crossing time (Tc) of protoplanetary systems with equal planetary masses and initial separation k 0 scaled by their mutual Hill radii.
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A simple framework for the derivation and analysis of effective one-step methods for ODEs
TL;DR: A simple framework to derive and analyse a class of one-step methods that may be conceived as a generalization of the class of Gauss methods, from which a large subclass of Runge–Kutta methods can be derived.
References
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Book
Mathematical aspects of classical and celestial mechanics
TL;DR: The main purpose of the book as discussed by the authors is to acquaint mathematicians, physicists and engineers with classical mechanics as a whole, in both its traditional and its contemporary aspects As such, it describes the fundamental principles, problems, and methods of classical mechanics, with the emphasis firmly laid on the working apparatus, rather than the physical foundations or applications.
Journal ArticleDOI
Triple collision in the collinear three-body problem.
TL;DR: This paper is concerned with Singularity (regularization) and regularization (singularity) in the 21st Century, specifically the aftermath of 9/11.
Journal ArticleDOI
The Analytical Foundations of Celestial Mechanics
TL;DR: Wintner as discussed by the authors discusses the need for a calendar from the beginning of civilization, namely the need to make itself felt, namely, for some form of calendar, which gives rise to the primitive study of astronomy, involving some knowledge of the sun and the moon.
Journal ArticleDOI
Poincare and the Three Body Problem.
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