Author's home page | Overview | Contents | Previous section | Next section 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.: click here to see their paper in 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). Author's home page | Overview | Contents | Previous section | Next section |
Collection of remarkable three-body motions – 3 of 7 |