Thesis (Ph. D.)--West Virginia University, 2008. / Title from document title page. Document formatted into pages; contains vi, 44 p. : ill. Includes abstract. Includes bibliographical references (p. 42-44).

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

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This thesis consists of two parts. In the first part, we study stability properties of Hamiltonian systems on the Wasserstein space. Let H be a Hamiltonian satisfying conditions imposed in the work of Ambrosio and Gangbo. We regularize H via Moreau-Yosida approximation to get H[subscript Tau] and denote by μ[subscript Tau] a solution of system with the new Hamiltonian H[subscript Tau] . Suppose H[subscript Tau] converges to H as τ tends to zero. We show μ[subscript Tau] converges to μ and μ is a solution of a Hamiltonian system which is corresponding to the Hamiltonian H. At the end of first part, we give a sufficient condition for the uniqueness of Hamiltonian systems. In the second part, we develop a general theory of differential forms on the Wasserstein space. Our main result is to prove an analogue of Green's theorem for 1-forms and show that every closed 1-form on the Wasserstein space is exact. If the Wasserstein space were a manifold in the classical sense, this result wouldn't be worthy of mention. Hence, the first cohomology group, in the sense of de Rham, vanishes.

A grid graph is a finite embedded subgraph of the infinite integer grid. A solid grid graph is a grid graph without holes, i.e., each bounded face of the graph is a unit square. The reconfiguration problem for Hamiltonian cycle or path in a sold grid graph G asks the following question: given two Hamiltonian cycles (or paths) of G, can we transform one cycle (or path) to the other using some "operation" such that we get a Hamiltonian cycle (or path) of G in the intermediate steps (i.e., after each application of the operation)? In this thesis, we investigate reconfiguration problems for Hamiltonian cycles and paths in the context of two types of solid graphs: rectangular grid graphs, which have a rectangular outer boundary, and L- shaped grid graphs, which have a single reflex corner on the outer boundary, under three operations we define, flip, transpose and switch, that are local in the grid. Reconfiguration of Hamiltonian cycles and paths in embedded grid graphs has potential applications in path planning, robot navigation, minimizing turn costs in milling problems, minimizing angle costs in TSP, additive manufacturing and 3D printing, and in polymer science. In this thesis, we introduce a complexity measure called bend complexity for Hamiltonian paths and cycles in grid graphs, and using those measures we measure complexity of a grid graph G and give upper and lower bounds on the maximum bend complexity of an mxn grid graph. We define three local operations, flip, transpose and switch, where local means that the operations are applied on vertices that are close in the grid graph but may not be close on the path or cycle. We show that any Hamiltonian cycle or path can be reconfigured to any other Hamiltonian cycle or path in an mxn rectangular grid graph, where m <= 4, using O(|G|) flips and transposes, regardless of the bend complexities of the two cycles. We give algorithms to reconfigure 1-complex Hamiltonian cycles in a rectangular or L-shaped grid graph G using O(|G|) flips and transposes, where the intermediate steps are also 1-complex Hamiltonian cycles. Finally, we establish the structure of 1-complex Hamiltonian paths between diagonally opposite corners s and t of a rectangular grid graph, and then provide a strategy, based on work in progress, for designing an algorithm to reconfigure between any two 1-complex s, t Hamiltonian paths using switch operations. / Graduate

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In this work, several problems in the field of Hamiltonian dynamics are studied. Chapter 1 is a short review of some basic results in the theory of Hamiltonian dynamics. In chapter 2, we study the problem of computing the geometric monodromy of the torus bundle defined by integrable Hamiltonian systems. We show that for two degree of freedom systems near an isolated critical value of the energy momentum map, the monodromy group can be determined solely from the local data of the energy momentum map at the singularity. Along the way, we develop a simple method for computing the monodromy group which covers all the known examples that exhibit nontrivial monodromy. In chapter 3, we consider the topological aspects of the Kirchhoff case of the motion of a symmetric rigid body in an infinite ideal fluid. The bifurcation diagrams are constructed and the topology of all the invariant sets are determined. We show that this system has monodromy. We show also that this system undergoes a Hamiltonian Hopf bifurcation as the couple resultant passes through a certain value when the steady rotation of the rigid body about its symmetry axis changes stability. Chapter 4 is devoted to checking Kolmogorov's condition for the square potential pendulum. We prove, by essentially elementary methods, that Kolmogorov's condition is satisfied for all of the regular values of the energy momentum map. In chapter 5, we use Ziglin's theorem to prove rigorously that some of the generalized two degree of freedom Toda lattices are non-integrable.

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