@MISC{Vanderbei_neworbits, author = {Robert J. Vanderbei}, title = {New Orbits for the n-Body Problem}, year = {} }

Share

OpenURL

Abstract

In this paper, we consider minimizing the action functional as a method for numerically discovering periodic solutions to the n-body problem. With this method, we can find a large number of choreographies and other more general solutions. We show that most of the solutions found, including all but one of the choreographies, are unstable. It appears to be much easier to find unstable solutions to the n-body problem than stable ones. Simpler solutions are more likely to be stable than exotic ones. 1. Least Action Principle Given n bodies, let mj denote the mass and zj(t) denote the position in R 2 = C of body j at time t. The action functional is a mapping from the space of all trajectories, z1(t), z2(t),..., zn(t), 0 ≤ t ≤ 2π, into the reals. It is defined as the integral over one period of the kinetic minus the potential energy: