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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.

Thesis (Ph.D.)--University of Delaware, 2006. / Principal faculty advisor: Sunil K. Agrawal, Dept. of Mechanical Engineering. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references (leaves 118-125). Available also in a digital version from Dissertation Abstracts.

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The dynamics of non-interacting, ultracold alkali atoms in the presence of counter-propagating lasers (optical lattice systems) is considered theoretically. The center of mass motion of an atom is such a system can be described by an effective Hamiltonian of a relatively simple form. Modulation of the laser fields implies a parametric variation of the effective Hamiltonian's eigenvalue spectrum, under which avoided crossings may occur. We investigate two dynamical processes arising from these near-degeneracies, which can be manipulated to coherently control atomic motion. First, we demonstrate the mechanism for the chaos-assisted, or multiple-state, tunneling observed in recent optical lattice experiments. Second, we propose a new method for the coherent acceleration of lattice atoms using the techniques of stimulated Raman adiabatic passage (STIRAP). In each case we use perturbation analysis to show the existence of a small, few level, subsystem of the full effective Schrödinger equation that determines the dynamics. / text

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The main principle of this thesis is to employ the geometric representation of Hamiltonian dynamics: in a broad sense, we study how to construct, in phase space, geometric structures that are related to a dynamical system. More specifically, we study the problem of constructing phase-space tori that are approximate invariant tori of a given Hamiltonian; also, using the constructed tori, we define an integrable Hamiltonian closely approximating the original one. The methods are generally applicable; as examples, we use gravitational potentials that are of interest in stellar dynamics. First, we construct tori for box and loop orbits in planar, barred potentials, thus demonstrating the applicability of the scheme to potentials that have more than one major orbit family. Also, we show that, in general, the construction scheme needs two types of canonical transformations together: point transformations as well as those expressed by generating functions. To complete the construction scheme, we show how to furnish the tori with consistent coordinate systems, i.e., how to recover the angle variables of a torus labelled by its actions. Next, the developed methods are employed in creating invariant phase-space tori in nonintegrable potentials supporting minor-orbit families. These tori are used to define an integrable Hamiltonian H₀, and a modified form of the standard Hamiltonian perturbation theory is then used to demonstrate that a minor-orbit family can be treated as one made up of orbits trapped by a resonance of H₀. Finally, we generalize the scheme further by constructing tori in time-reversal asymmetric Hamiltonians (by considering the motion in a rotating frame of reference), and study the transition from locally contained stochasticity to global chaos. Using both near-integrable 'laboratory' Hamiltonians and those for which we construct tori, we investigate the transition in the light of the resonance overlap criterion.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.