Publication Date
Summer 2015
Degree Type
Thesis
Degree Name
Master of Science (MS)
Department
Mathematics
Advisor
Slobodan Simic
Keywords
hamiltonian system, kam theory, periodicity, perturbation theory, symplectic integrator, variational integrator
Subject Areas
Mathematics
Abstract
Classical numerical integrators do not preserve symplecticity, a structure inherent in Hamiltonian systems. Thus, the trajectories they produce cannot be expected to possess the same qualitative behavior observed in the original system. Pooling recent results from O'Neale and West, we explore a particular class of numerical integrators, the variational integrator, that preserves one aspect of the range of behavior present in Hamiltonian systems, the periodicity of trajectories. We first establish the prerequisites and some key concepts from Hamiltonian systems, particularly symplecticity and action-angle coordinates. Through perturbation theory and its complications manifested in small divisor problems, we motivate the necessity for KAM theory. With O'Neale's KAM-type theorem, we observe the preservation of periodicity by symplectic one-step methods. Lastly, we show that the variational integrator introduced by West possesses the defining characteristics of symplectic one-step methods, and therefore also preserves periodicity of the original trajectories.
Recommended Citation
Liu, Jian-Long, "Preservation of Periodicity in Variational Integrators" (2015). Master's Theses. 4596.
DOI: https://doi.org/10.31979/etd.c4z9-y9yp
https://scholarworks.sjsu.edu/etd_theses/4596