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DSpace at Massey University author O'Neale, Dion Robert James 2010-01-05T22:33:09Z NO_RESTRICTION2010-01-05T22:33:09Z 2009 http://hdl.handle.net/10179/1127 Q111965394 https://www.wikidata.org/wiki/Q111965394 This thesis concerns the study of geometric numerical integrators and how they preserve phase space structures of Hamiltonian ordinary differential equations. We examine the invariant sets of differential equations and investigate which numerical integrators preserve these sets, and under what conditions. We prove that when periodic orbits of Hamiltonian differential equations are discretized by a symplectic integrator they are preserved in the numerical solution when the integrator step size is not resonant with the frequency of the periodic orbit. The preservation of periodic orbits is the result of a more general theorem which proves preservation of lower dimensional invariant tori from dimension zero (fixed points) up to full dimension (the same as the number of degrees of freedom for the differential equation). The proof involves first embedding the numerical trajectory in a non-autonomous flow and then applying a KAM type theorem for flows to achieve the result. This avoids having to prove a KAM type theorem directly for the symplectic map which is generally difficult to do. We also numerically investigate the break up of periodic orbits when the integrator's step size is resonant with the frequency of the orbit. We study the performance of trigonometric integrators applied to highly oscillatory Hamiltonian differential equations with constant frequency. We show that such integrators may not be as practical as was first thought since they suffer from higher order resonances and can perform poorly at preserving various properties of the di fferential equation. We show that, despite not being intended for such systems, the midpoint rule performs no worse than many of the trigonometric integrators, and indeed, better than some. Lastly, we present a numerical study of a Hamiltonian system consisting of two magnetic moments in an applied magnetic field. We investigate the effect of both the choice of integrator and the choice of coordinate system on the numerical solutions of the system. We show that by a good choice of integrator (in this case the generalised leapfrog method) one can preserve phase space structures of the system without having to resort to a change of coordinates that introduce a coordinate singularity. en The Author Geometric integrator Differential equations Preservation of phase space structure in symplectic integration : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand Thesis 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 URL https://mro.massey.ac.nz/bitstreams/ca1b8aa4-ad92-414e-beff-43ebdabea2c1/download File MD5 3c33479db4292330d4384c8fa95f27af 29027254 application/pdf 02_whole.pdf URL https://mro.massey.ac.nz/bitstreams/88c730b3-db79-41b4-85dd-a2f083409ea8/download File MD5 6baae05ac7fc1bf2c1a7fdae479d3de0 127854 application/pdf 01_front.pdf URL https://mro.massey.ac.nz/bitstreams/1cbe344a-c379-4efd-9deb-f57d20513165/download File MD5 674b17f4ecd18d013249fc799acb3d26 15505 text/plain 01_front.pdf.txt URL https://mro.massey.ac.nz/bitstreams/e3ea2ac7-5e05-4fe8-8253-f0a6de2e1e39/download File MD5 18ff326b8f6046e75e268037b265b846 361030 text/plain 02_whole.pdf.txt