Date of Award
1-1-2023
Document Type
Dissertation
Degree Name
Ph.D. in Mathematics
First Advisor
Samuel T. Lisi
Second Advisor
Bing Wei
Third Advisor
Haidong Wu
Relational Format
dissertation/thesis
Abstract
A Hamiltonian system is a dynamical system governed by Hamilton’s equations. Hamiltonian systems provide a framework for studying classical mechanics. One fundamental principle in Hamiltonian mechanics is the conservation of energy along the flow of the Hamiltonian vector field. The smooth hypersurface of a Hamiltonian function on a symplectic manifold has a contact structure, and the associated Reeb vector field is, up to a scale, the Hamiltonian vector field. We are interested in the non-smooth case and a symplectic polytope makes a good model to study.
In this dissertation, we study the Reeb dynamics on the boundary of a symplectic polytope. In particular, we calculate the Conley-Zehnder index of a path of symplectic matrices starting from the identity to a symplectic matrix that does not have eigenvalue 1. We start by analyzing the mapping torus, and study the Conley-Zehnder index in that context. We then analyze the situation relevant for 4-dimensional symplectic polytope in Chaidez and Hutchings’ research where the quaternionic trivialization plays an essential role. The new part of this work is that we introduce a family of trivializations that generalizes the quaternions in R4. We prove our trivializations give positive elliptic transition matrices and the same Conley-Zehnder index in comparison to the quaternionic trivialization. We generalize the transition matrices for symplectic polytopes in dimension 6 and higher.
Recommended Citation
Yu, Yutian, "A Discrete Conley-Zehnder Index for Symplectic Polytopes" (2023). Electronic Theses and Dissertations. 2779.
https://egrove.olemiss.edu/etd/2779