Symmetric Regularization, Reduction and Blow-Up of the Planar Three-Body Problem

We carry out a sequence of coordinate changes for the planar three-body problem which successively eliminate the translation and rotation symmetries, regularize all three double collision singularities and blow-up the triple collision. Parametrizing the configurations by the three relative position vectors maintains the symmetry among the masses and simplifies the regularization of binary collisions. Using size and shape coordinates facilitates the reduction by rotations and the blow-up of triple collision while emphasizing the role of the shape sphere. By using homogeneous coordinates to describe Hamiltonian systems whose configurations spaces are spheres or projective spaces, we are able to take a modern, global approach to these familiar problems. We also show how to obtain the reduced and regularized differential equations in several convenient local coordinates systems.Comment: 51 pages, 4 figure

Existence of either a periodic collisional orbit or infinitely many consecutive collision orbits in the planar circular restricted three-body problem

In the restricted three-body problem, consecutive collision orbits are those orbits which start and end at collisions with one of the primaries. Interests for such orbits arise not only from mathematics but also from various engineering problems. In this article, using Floer homology, we show that there are either a periodic collisional orbit, or infinitely many consecutive collision orbits in the planar circular restricted three-body problem on each bounded component of the energy hypersurface for Jacobi energy below the first critical value.Comment: 13 p

The contact geometry of the restricted 3-body problem

We show that the planar circular restricted three body problem is of restricted contact type for all energies below the first critical value (action of the first Lagrange point) and for energies slightly above it. This opens up the possibility of using the technology of Contact Topology to understand this particular dynamical system.Comment: 29 pages, 1 figur

Dynamical convexity of the Euler problem of two fixed centers

We give thorough analysis for the rotation functions of the critical orbits from which one can understand bifurcations of periodic orbits. Moreover, we give explicit formulas of the Conley-Zehnder indices of the interior and exterior collision orbits and show that the universal cover of the regularized energy hypersurface of the Euler problem is dynamically convex for energies below the critical Jacobi energy.Comment: Final version, title changed, to appear in Math. Proc. Cambridge Philos. So

A construction of lattice chiral gauge theories

Path integration over Euclidean chiral fermions is replaced by the quantum mechanics of an auxiliary system of non--interacting fermions. Our construction avoids the no--go theorem and faithfully maintains all the known important features of chiral fermions, including the violation of some perturbative conservation laws by gauge field configurations of non--trivial topology.Comment: 74 pages, uuencoded, gz compressed PS fil