Global Aspects of the N-Body Problem

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The Newtonian N-body problem suffers (or enjoys) two difficulties not shared by the typical dynamical systems of general theory: collision singularities which render the flow incomplete, and symmetries which make all its periodic orbits degenerate. The investigator and his colleague eliminate these two difficulties. They eliminate collision singularities by Levi-Civita regularization, Kuustaanheimo-Steifel regularization or McGehee blow-up. They eliminate the symmetries by symplectic reduction and invariant theory. They call the resulting regularized reduced flow the "global regularization." This flow is complete and has the minimum number of degrees of freedom possible among for a flow encoding the N-body problem. Given the global regularization, the investigator and his colleague study previously inaccessible problems within the N-body problem, such as the dynamical and variational nature of collision-collision solutions, and the existence or lack of existence of a symbolic dynamics for the planar zero-angular momentum three-body problem whose 3 symbols are the 3 types of syzygies (collinearities). Imagine the moon going around the earth, the earth going around the sun, and the sun spinning around the galaxy. The dominant force in the large scale dynamics of the universe is the attractive force of gravity and is responsible for the large scale rotational type motions of celestial bodies. The N-body equations are the equations governing this motion. ("N" stands for the number of bodies.) When two bodies get close the forces get large to the point that when the bodies collide the forces become infinity. This infinity is called a "collision singularity." If the whole universe is translated, or rotated some fixed amount, the motion remains identical and so there are extra variables in the usual N-body equations, namely the variables describing the "origin" and "axes" of the universe. The investigator and his collaborator eliminate the collision singularities, and the extra variables in the N-body equations. These eliminations yield a potentially more useful system of equations for the motion, a system free of infinities which may provide a good platform from which to solve some of the many open problems about one of the dominant dynamical behaviors going on in our universe.