The N-body problem is defined and its equations of motion are presented. A new time transformation based on two-body problem series solutions is introduced and used to develop a time-dependent, analytic, first-order solution of the N-body problem. A process of deriving higher-order, time-dependent, analytic solutions of the N-body problem is also introduced. These analytic solutions are used to investigate periodic and non-periodic trajectories in the restricted three-body problem and are shown to be capable of describing complete, N-body trajectories with good accuracy and with far fewer function evaluations than are required by numerical integration.
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