### Computer Presentations

The links below allow you to download copies of computer presentations of some talks I have given. There are versions for Keynote and PDF. Unfortunately, the PDF versions will not show the movies (so get a Mac !).

Finding Brake Orbits. Talk given in Mexico City, December 2010. Describes an existence proof for simple, periodic orbits of the isosceles three-body problem which begin with zero initial velocity (i.e., they are periodic brake orbits)

Regularized Three-Body Problem. Talk given in Lyon, France June 2012. Describes the geometry of the reduced and regularized planar three-body problem and some interesting approaches to the various coordinate changes.

Chaos in the Three-Body Problem. Talk given in Paris, France, November 2012 at the Poincare 100 conference at the Institut Henri Poincare. Poincare's famous quote about chaotic tangles of stable and unstable manifolds is illustrated with computer pictures of the restricted three-body problem. Then further developments about chaos near infinity and near triple collision in the Sitnikov problem are illustrated.

Blowing Up the N-Body Problem. 4 hour course given in Asiago, Italy in June 2018. Lecture 1 covers McGehee's blowup method for studying total collision and totally parabolic motions and uses the two-body problem as an example. Lecture 2
moves on to the planar three-body problem, introduces the shape sphere and studies the stable and unstable manifolds of the equilibrium points. Lecture 3 covers
symbolic dynamics and chaos in the planar three-body problem. Lecture 4 covers syzygy sequences and more about parabolic infinity.

Relative Equilibria of Gravitationally Interacting Rigid Bodies. Lecture in Venice, Italy, June 2018. It introduces relative equilibria as possible end states of tidal evolution. The relative equilibria are critical points of energy on manifolds of fixed momentum and angular momentum. The minimal ones are the likely end states of tidal evolution. Part 1 shows that for three or more bodies, minima do not exist -- all of the critical points are saddles. Part 2 describes lower bounds for the number of relative equilibria for two bodies using Morse theory and LS category.