Persistent Link:
http://hdl.handle.net/10150/185996
Title:
Geometry of two degree of freedom integrable Hamiltonian systems.
Author:
Zou, Maorong.
Issue Date:
1992
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
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.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Hamiltonian systems.; Monodromy groups.; Topology.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Mathematics; Graduate College
Degree Grantor:
University of Arizona
Committee Chair:
Flaschka, Hermann

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleGeometry of two degree of freedom integrable Hamiltonian systems.en_US
dc.creatorZou, Maorong.en_US
dc.contributor.authorZou, Maorong.en_US
dc.date.issued1992en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.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.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectHamiltonian systems.en_US
dc.subjectMonodromy groups.en_US
dc.subjectTopology.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.chairFlaschka, Hermannen_US
dc.contributor.committeememberErcolani, Nicholasen_US
dc.contributor.committeememberYoung, Lai-Sangen_US
dc.identifier.proquest9307660en_US
dc.identifier.oclc703161677en_US
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