A New Set of Variables in the Three-Body Problem

Details A New Set of Variables in the Three-Body Problem A New Set of Variables in the Three-Body Problem

Feb 2010

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2010pasj...62....1k&link_type=abstract

Publications of the Astronomical Society of Japan, Vol.62, No.1, pp.1--7

Astronomy and Astrophysics

Astronomy

1

Three-Body Problem, Shape Spheres

Scientific paper

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, Λ, Θ, k, ω), or equivalently (λ, θ, L, dot{I}, k, ω) for the two-dimensional problem, and (λ, θ, L, dot{I}, k, ω, φ, ψ) for the three-dimensional problem. Here, (λ, θ) and (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, dot{I} is the time derivative of the moment of inertia, and ω, φ, and ψ 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 (λ, θ, k = 0, L = 0, dot{I} = 0, ω = 0). We show that the hyper-surface dot{I} = 0 is a global surface of section.

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