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Symbolic Dynamics and the Braid Group $\pi_1({\cal S})$

Recent work of Montgomery1 has made much use of the topology of the shape sphere ${\cal S}$, which is topologically $S^2 \setminus \{*,*,*\}$. Specifically, for suitable homotopy classes in the colored braid group $\pi_1({\cal S})$ there is an action-minimizing periodic orbit of the 3-body problem whose image on the shape sphere realizes the homotopy class.

The free (unbased) homotopy classes are encoded by the order and orientation with which the projection onto the shape sphere winds around the excluded collision points. This encoding also provides a kind of symbolic dynamics for the three-body problem as follows.

The braid group (i.e. the fundamental group of the shape sphere) is a free group on six generators $A_\pm$, $B_\pm$, $C_\pm$ with the following rules:

The equator is divided into three arcs $A, B, C$ by the removal of the collision points. A non-trivial (i.e. not contractible) homotopy class in the braid group must wind around the punctures on the equator. Which arc is crossed is encoded by its letter; the direction in which it is crossed is encoded by a plus or minus sign. Thus the rules are natural: the first specifies that a trajectory cannot cross the equator twice successively in the same direction, and the second removes as homotopically trivial those pairs of crossings which do not wind around a puncture.

Each trajectory of the three-body problem uniquely defines a path on the shape sphere and hence an encoding of a word in the braid group. Montgomery (1998) showed that under suitable hypotheses, the opposite is also true: for each word in the braid group there is a trajectory of the three-body problem which minimizes the action integral (and hence is realizable).


next up previous
Next: The Chenciner-Montgomery Figure-Eight Up: The Shape Sphere Previous: Coordinates on the Shape
Matthew Salomone 2003-07-24