2007/03/26 by Kenji Hiro Kuwabara, Kuwabara, Kenji Hiro, Kiyotaka Tanikawa +1
Engineering · Physics and Astronomy · #Astro and Planetary Science #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Space Satellite Systems and Control #Spacecraft Dynamics and Control #nlin.CD
paper · pdf · doi:10.48550/arxiv.nlin/0703052
9 pages, 3 figures
arxiv created 2007/03/26 · openalex publication_date 2007/03/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose a set of variables of the general three-body problem both for two-dimensional and three-dimensional cases. Variables are (λ,θ,Λ, Θ,k,ω) or equivalently (λ,θ,L,I,k,ω) for the two-dimensional problem, and (λ,θ,L,I,k,ω,ϕ,ψ) for the three-dimensional problem. Here (λ,θ) and (Λ,Θ) specifies the positions in the shape spheres in the configuration and momentum spaces, k is the virial ratio, L is the total angular momentum, 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 \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 (λ,θ,k=0,L=0,I=0,ω=0). We show that the hyper-surface I = 0 is a global surface of section.