Application of classical non-linear Liouville dynamic approximations

This dissertation examines the application of the Liouville operator to problems in classical mechanics. An approximation scheme or methodology is sought that would allow the calculation of the position and momentum of an object at a specified later time, given the initial values of the object's position and momentum at some specified earlier time. The approximation scheme utilizes matrix techniques to represent the Liouville operator.

An approximation scheme using the Liouville operator is formulated and applied to several simple one-dimensional physical problems, whose solution is obtainable in terms of known analytic functions. The scheme is shown to be extendable relative to cross products and powers of the variables involved. The approximation scheme is applied to a more complicated one-dimensional problem, a quartic perturbed simple harmonic oscillator, whose solution is not capable of being expressed in terms of simple analytic functions. Data produced by the application of the approximation scheme to the perturbed quartic harmonic oscillator is analyzed statistically and graphically. The scheme is reapplied to the solution of the same problem with the incorporation of a drag term, and the results analyzed. The scheme is then applied to a simple physical pendulum having a functionalized potential in order to ascertain the limits of the approximation technique. The approximation scheme is next applied to a two-dimensional non-perturbed Kepler problem. The data produced is analyzed statistically and graphically. Conclusions are drawn and suggestions are made in order to continue the research in several of the areas presented.