Three Body Problem

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S.p. Sosnitskii - One of the best experts on this subject based on the ideXlab platform.

Kiyotaka Tanikawa - One of the best experts on this subject based on the ideXlab platform.

  • The Three-Body Problem from Pythagoras to Hawking
    2016
    Co-Authors: Mauri Valtonen, Joanna Anosova, Konstantin Kholshevnikov, Aleksandr Mylläri, Victor Orlov, Kiyotaka Tanikawa
    Abstract:

    This book, written for a general readership, reviews and explains the Three-Body Problem in historical context reaching to latest developments in computational physics and gravitation theory. The Three-Body Problem is one of the oldest Problems in science and it is most relevant even in today’s physics and astronomy. The long history of the Problem from Pythagoras to Hawking parallels the evolution of ideas about our physical universe, with a particular emphasis on understanding gravity and how it operates between astronomical bodies. The oldest astronomical Three-Body Problem is the question how and when the moon and the sun line up with the earth to produce eclipses. Once the universal gravitation was discovered by Newton, it became immediately a Problem to understand why these Three-bodies form a stable system, in spite of the pull exerted from one to the other. In fact, it was a big question whether this system is stable at all in the long run. Leading mathematicians attacked this Problem over more than two centuries without arriving at a definite answer. The introduction of computers in the last half-a-century has revolutionized the study; now many answers have been found while new questions about the Three-Body Problem have sprung up. One of the most recent developments has been in the treatment of the Problem in Einstein’s General Relativity, the new theory of gravitation which is an improvement on Newton’s theory. Now it is possible to solve the Problem for Three black holes and to test one of the most fundamental theorems of black hole physics, the no-hair theorem, due to Hawking and his co-workers

  • A New Set of Variables in the Three-Body Problem
    Publications of the Astronomical Society of Japan, 2010
    Co-Authors: Kenji Hiro Kuwabara, Kiyotaka Tanikawa
    Abstract:

    We propose a set of variables of the general Three-Body Problem both for two-dimensional and Three-dimensional cases. The variables are (� ,� , Λ, Θ,k,!), or equivalently (� ,� ,L, P I;k;!/for the two-dimensional Problem, and (� ,� ,L, P I ,k,!,� , ) for the Three-dimensional Problem. Here, (� ,� )a nd (Λ, Θ/ specify the positions in the shape spheres in the configuration and momentum spaces, k is the virial ratio, L is the total angular momentum, P I is the time derivative of the moment of inertia, and!,� ,a nd are the Euler angles to bring the momentum triangle from the nominal position to a given position. This set of variables defines a shape space of the Three-Body Problem. This is also used as an initial-condition space. The initial condition of the so-called free-fall Three-Body Problem is (� ,� , k = 0,L = 0, P I = 0,! = 0). We show that the hyper-surface P I = 0 is a global surface of section.

  • A trial symbolic dynamics of the planar Three-Body Problem
    arXiv: Chaotic Dynamics, 2008
    Co-Authors: Kiyotaka Tanikawa, Seppo Mikkola
    Abstract:

    Symbolic dynamics is applied to the planar Three-Body Problem. Symbols are defined on the planar orbit when it experiences a syzygy crossing. If the Body i is in the middle at the syzygy crossing and the vectorial area of the triangle made with Three bodies changes sign from + to -, number i is given to this event, whereas if the vectorial area changes sign from - to +, number i+3 is given. We examine the case of free-fall Three-Body Problem for the first few digits of symbol sequences, and we examine the case with angular momentum only for the first digit of the symbol sequences. This trial experiments show some new aspects of the planar Three-Body Problem.

  • A new set of variables in the Three-Body Problem
    arXiv: Chaotic Dynamics, 2007
    Co-Authors: Kenji Hiro Kuwabara, Kiyotaka Tanikawa
    Abstract:

    We propose a set of variables of the general Three-Body Problem both for two-dimensional and Three-dimensional cases. Variables are $(\lambda,\theta,\Lambda, \Theta,k,\omega)$ or equivalently $(\lambda,\theta,L,\dot{I},k,\omega)$ for the two-dimensional Problem, and $(\lambda,\theta,L,\dot{I},k,\omega,\phi,\psi)$ for the Three-dimensional Problem. Here $(\lambda,\theta)$ and $(\Lambda,\Theta)$ specifies the positions in the shape spheres in the configuration and momentum spaces, $k$ is the virial ratio, $L$ is the total angular momentum, $\dot{I}$ is the time derivative of the moment of inertia, and $\omega,\phi$, and $\psi$ are the Euler angles to bring the momentum triangle from the nominal position to a given position. This set of variables defines a {\it shape space} of the Three-Body Problem. This is also used as an initial condition space. The initial condition of the so-called free-fall Three-Body Problem is $(\lambda,\theta,k=0,L=0,\dot{I}=0,\omega=0)$. We show that the hyper-surface $\dot{I} = 0$ is a global surface of section.

  • Oscillatory orbits in the planar ThreeBody Problem with equal masses
    Celestial Mechanics and Dynamical Astronomy, 1998
    Co-Authors: Kiyotaka Tanikawa, Hiroaki Umehara
    Abstract:

    In the free‐fall ThreeBody Problem, distributions of escape, binary, and triple collision orbits are obtained. Interpretation of the results leads us to the existence of oscillatory orbits in the planar ThreeBody Problem with equal masses. A scenario to prove their existence is described.

Richard Montgomery - One of the best experts on this subject based on the ideXlab platform.

  • Constructing the Hyperbolic Plane as the reduction of a Three-Body Problem
    arXiv: Dynamical Systems, 2016
    Co-Authors: Richard Montgomery
    Abstract:

    We construct the hyperbolic plane with its geodesic flow as the scale plus symmetry reduction of a Three-Body Problem in the Euclidean plane. The potential is $-I/\Delta^2$ where $I$ is the triangle's moment of inertia and $\Delta$ its area. The reduction method uses the Jacobi-Maupertuis metric, following the author's earlier paper "Putting Hyperbolic Pants on a Three-Body Problem".

  • The Three-Body Problem and the Shape Sphere
    The American Mathematical Monthly, 2015
    Co-Authors: Richard Montgomery
    Abstract:

    The Three-Body Problem defines dynamics on the space of triangles in the plane. The shape sphere is the moduli space of oriented similarity classes of planar triangles and lies inside shape space, a Euclidean Three-space parameterizing oriented congruence classes of triangles. We derive and investigate the geometry and dynamics induced on these spaces by the Three-Body Problem. We present two theorems concerning the Three-Body Problem whose discovery was made through the shape space perspective. 1. INTRODUCTION. In 1667 Newton (21) posed the Three-Body Problem. Central questions within the Problem remain open today despite penetrating work over the centuries by many of our most celebrated mathematicians, including Euler, Lagrange, Laplace, Legendre, d'Alembert, Clairaut, Delanay, Poincare, Birkhoff, Seigel, Kol- mogorov, Arnol'd, Moser, and Smale. The Problem, in its crudest form, asks us to solve the system of Ordinary Differ- ential Equations (ODEs), Equations (1), governing the motion of Three point masses moving in space under the sole influence of each other's mutual gravitational attrac- tions. The masses form the vertices of a triangle, so we can think of the Problem as one of moving triangles. According to the relativity principle of Galilieo, the laws of physics are invariant under isometries. Isometries are the congruences of Euclid. Two triangles are congruent if and only if the lengths of their Three sides are equal. Is there a system of second order ODEs in the lengths of the Three sides that describes the Three-Body Problem? All attempts to write down such a system of ODEs break down at collinear triangles. Instead, we will derive Three alternative variables to the side lengths of a triangle and show that there is such a system of ODEs in these variables. Unlike the side lengths, the alternative variables are not invariant under all congruences but, rather, only un- der "oriented congruences." Two triangles are "oriented congruent" if a composition of a translation and rotation takes one to the other. Oriented congruence excludes re- flections. We define shape space to be the space of oriented congruence classes of planar triangles. Shape space is homeomorphic to R 3 and is parameterized by the vec- tor (w1 ,w 2 ,w 3) formed by these Three alternative variables. We derive second order ODEs (Equation (54)) for these alternative variables that encode a special case of the Three-Body Problem. Shape space is homeomorphic to R 3 but it is not isometric to R 3 . The shape space metric is not Euclidean. Nevertheless the shape space metric does enjoy spherical sym- metry, which means that at the heart of shape space geometry is a two-dimensional sphere. We call this sphere the shape sphere. Its points represent oriented similarity classes of planar triangles (Figure 2). The purpose of this article is to describe the ge- ometry of this sphere, how it relates to the Three-Body Problem, and how this relation yields new insights into this age-old Problem.

  • The Three-Body Problem and the shape sphere
    arXiv: Dynamical Systems, 2014
    Co-Authors: Richard Montgomery
    Abstract:

    [This is an expository article. I have submitted it to the American Mathematical Monthly.] The Three-Body Problem defines a dynamics on the space of triangles in the plane. The shape sphere is the moduli space of oriented similarity classes of planar triangles and lies inside shape space, a Euclidean 3-space parametrizing oriented congruence classes of triangles. We derive and investigate the geometry and dynamics induced on these spaces by the Three-Body Problem. We present two theorems concerning the Three-Body Problem whose discovery was made through the shape space perspective

T. Van Bemmelen - One of the best experts on this subject based on the ideXlab platform.

Claudia Tamayo - One of the best experts on this subject based on the ideXlab platform.