The shape sphere: a new vista on the three body problem (David Alcaraz conference: Video conference)

CDSNS Colloquium
Tuesday, April 12, 2016 - 13:30
1 hour (actually 50 minutes)
Skiles 005
Univ. California Santa Cruz
Video Conference David Alcaraz confernce.  Newton's famous    three-body problem defines dynamics on the space of congruence classes of triangles in the plane.  This space is a three-dimensional non-Euclidean  rotationally symmetric metric space ``centered'' on  the  shape sphere. The shape sphere is  a two-dimensional sphere whose points represent   oriented similarity classes of planar triangles. We describe how the sphere arises from the three-body problem and  encodes its dynamics.    We will  see how   the classical solutions of Euler and Lagrange, and the relatively recent figure 8 solution are encoded as points or curves on  the sphere.  Time permitting, we will show how the sphere pushes us to formulate natural topological-geometric questions about three-body solutions and helps supply the answer to some of these questions.  We may take a brief foray into the planar N-body problem and  its  associated ``shape sphere'' :   complex projective N-2 space.