The homotopy class of this path is a strong topological invariant of the system. In a Lagrangian system, the three bodies always form an equilateral triangle, although it may grow, shrink, or rotate. Thus Lagrangian systems map to 0 or to infinity, the two poles of the Riemann sphere. One gets a feel for the complexity of a system by constructing its path traced out on the Riemann sphere by the triangle the bodies bound. Positively oriented triangles map outside the unit circle, negatively oriented ones inside, triangles with the three vertices collinear map to the circle itself.
Isosceles triangles map to one of three lines running through the origin and a , b , or c .
The principal result of Chenciner and Montgomery is that in the homotopy class of the track of the path exhibited above there does exist the track of a three-body system . The three-body problem 2. The figure-eight 3. The triangle construction 4. The search for exotic orbits 6.