Physics – Mathematical Physics
Scientific paper
2004-04-24
J. Phys. A: Math. Gen. 37 (2004) 10571-10584
Physics
Mathematical Physics
14 pages, 3 figures, submitted to The Journal of Physics A
Scientific paper
10.1088/0305-4470/37/44/008
Geometrical properties of three-body orbits with zero angular momentum are investigated. If the moment of inertia is also constant along the orbit, the triangle whose vertexes are the positions of the bodies, and the triangle whose perimeters are the momenta of the bodies, are always similar (``synchronised similar triangles''). This similarity yields kinematic equalities between mutual distances and magnitude of momenta. Moreover, if the orbit is a solution to the equation of motion under homogeneous potential, the orbit has a new constant involving momenta. For orbits with zero angular momentum and non-constant moment of inertia, we introduce scaled variables, positions divided by square root of the moment of inertia and momenta derived from the velocity of the scaled positions. Then the similarity and the kinematic equalities hold for the scaled variables. Using this similarity, we prove that any bounded three-body orbit with zero angular momentum under homogeneous potential whose degree is smaller than 2 has infinitely many collinear configurations (syzygies or eclipses) or collisions.
Fujiwara Toshiaki
Fukuda Hiroshi
Kameyama Atsushi
Ozaki Hiroshi
Yamada Michio
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