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3. Figure-eight periodic planar three-body motion

Three bodies of equal masses can execute a surprisingly simple periodic planar motion chasing each other along a highly symmetric figure-8 closed orbit. This dynamically stable motion was probably first mentioned by C. Moore (Braids in Classical Dynamics, Phys. Rev. Lett. 70, 3675 3679, 1993) and described recently in detail by A. Chenciner and R. Montgomery (A remarkable periodic solution of the three body problem in the case of equal masses. Annals of Mathematics, 152 (2000), 881 901 ( click here to see their paper on the Web).

The motion is characterized by zero angular momentum and a very rich symmetry pattern. Starting from the collinear configuration, the bodies exactly reproduce this configuration after a certain time interval T, during which each body traces one and the same closed figure-8 curve. After T/3 time interval (one-third of the total period), the bodies again occur on the same straight line as in the initial configuration, but in a different order. There is one more (symmetric) straight line on which the bodies simultaneously occur in T/6 time interval after the first collinear configuration. In the meantime (after T/12) the bodies occur in an isosceles triangular configuration.

Next section shows this motion simultaneously in two reference frames (center-of-mass frame and the frame of one of the bodies).

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Collection of remarkable three-body motions 3 of 7