Geometry of two degree of freedom integrable Hamiltonian systems.
Committee ChairFlaschka, Hermann
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PublisherThe University of Arizona.
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AbstractIn 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.