| The Principle of Least Action provides an approach to the search for periodic solutions of the Newtonian three-body problem. Using variational techniques, we consider Meyer's three-body "comet" situation as presented his Periodic Solutions of the N-Body Problem. These orbits have two bodies close together and a third body on a distant orbit. This is the typical situation for triple star systems. The classical approach to show existence of periodic solutions relies on a limiting argument and the implicit function theorem. It can only be applied to solutions for which the primary bodies are very close to one other.; We look at curves with three bodies of equal mass that have collinear initial position and, after a given time, end up in an isosceles configuration with a fixed amount of rotation. When viewed on the shape sphere, the orbit leaves the equator with a transverse intersection and, in the given time, intersects with a particular meridian corresponding to isosceles configurations. Over this family of curves we minimize the action. In order to show the minimum exists, we show compactness properties of the space of functions and coercivity of the action integral. Since the variational arguments are more global in nature than limiting arguments, we find a family of periodic orbits extending from the extreme "comet" case to orbits where the "comet" passes close to both primaries. All of these orbits have the same topology and they can be deformed into each other without passing through collision. Using the method of steepest descent, we find approximate initial conditions of these orbits. |
