Numerical Methods for Hamiltonian Systems: Theory, Numerics and Case Studies

Open Access
Author:
Zhan, Hongyuan
Area of Honors:
Mathematics
Degree:
Bachelor of Science
Document Type:
Thesis
Thesis Supervisors:
  • Xiantao Li, Thesis Supervisor
  • Victoria V Sadovskaya, Honors Advisor
Keywords:
  • numerical methods
  • Hamiltonian system
Abstract:
This thesis consists of my studies of Hamiltonian systems during the Applied Math REU program at the Department of Mathematics, Penn State, and the follow-up research. The study focuses on three aspects primarily, as indicated in the chapter names: theory, numerical methods and solid examples. These chapters follow a logical sequence. We first examine the qualitative properties of Hamiltonian systems, including the derivation of the Hamiltonian and Lagrangian formulations and proving the symplecticity of flows. Next we present different numerical methods and classify the methods according to whether they preserve some important qualitative properties mentioned in the previous chapter. In the final chapter, after developing the theories and numerical methods, we use these tools to investigate a particular Hamiltonian system, the flexible pendulum model. We analyze some special properties of this model and use numerical simulations to confirm the analytical results. The numerical results also serve as examples of findings in Chapter two.