The Principle of Stationary Action and Numerical Methods for N-Body Problems*

Two-point boundary-value problems for conservative systems are studied in the context of the stationary action principle. In particular, we consider the case where the initial boundary condition is the system position, and the terminal boundary condition may be a combination of position and velocity data. The emphasis is on the N -body problem under gravitation. When the duration is sufficiently short, one may use a differential game formulation to obtain a fundamental solution, where for specific initial position and terminal data, one obtains the particular solution via a min-plus convolution of a function related to the terminal data and another function associated with the fundamental solution. That latter function is obtained by minimization of a parameterized linear functional over a convex set. This convex set is the fundamental solution. For longer duration problems, one takes a stationary point rather than a minimum.