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Determination of motion from orbit in the three-body problem

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  • We discuss the equal mass three-body motion in which the shape of the orbit is given. The conservation of the center of mass and a constant of motion (the total angular momentum or the total energy) leads to the uniqueness of the equal mass three-body motion in given some sorts of orbits. Although the proof was already published on an article by the present authors in 2009, here we give some complementary explanations. We show that, even in the unequal mass three-body periodic motions in which each of bodies draws its own orbit, the shape of the orbits, conservation of the center of mass and a constant of motion provide some candidates of the motion of three bodies. The reality of the motion should be tested whether the equation of motion is satisfied or not. Even if the three bodies draw unclosed orbits, we can show that similarly.
    Mathematics Subject Classification: Primary: 70F07, 70F15; Secondary: 51P05.

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