Solution and Characterization Methods

Abstract

In the following we discuss various methods by which the equations of motion can be solved to find the trajectory of a particle or a solution flow in the neighborhood of a trajectory. We make an implicit assumption that the equations we wish to solve are non-integrable, meaning that there exists no closed-from solution. In many situations we will rely on a closed-form solution of a simpler, non-perturbed system to help generate a solution to a more general problem. As the given problems are non-integrable, we are always limited in that our solutions and solution methods are approximate, and only represent the true solution up to some degree of precision over a finite timespan. Some exceptions to these occur for the special solutions of dynamical systems, namely (relative) equilibria and periodic orbits. A relative equilibria is an exact solution to a system, and in general does not rely on a solution of the simpler, non-perturbed problem. A periodic orbit is not exact, as it has a limit on the precision (except in the rare cases where a completely convergent series is found in closed form), although there is no limit on the timespan for the solution, that being infinite. All other solutions have limitations on both precision and time.