Use this URL to cite or link to this record in EThOS:
Title: Classical and quantum integrability in dimensions two and three
Author: Ye, Yiru
ISNI:       0000 0004 9346 7477
Awarding Body: Loughborough University
Current Institution: Loughborough University
Date of Award: 2019
Availability of Full Text:
Access from EThOS:
Access from Institution:
The thesis consists of two parts. In the first part, we study the integrability of geodesic flows on 3-manifolds that admit SL(2,R) geometry in Thurston's sense. The main examples are the quotients M3/Γ = Γ\P SL(2,R), where Γ ⊂ P SL(2, R) is a cofinite Fuchsian group. We show that the corresponding phase space T *M3/Γ contains two open regions with integrable and chaotic behaviour. In the integrable region we have Liouville integrability with analytic integrals, while in the chaotic region the system is not Liouville integrable even in smooth category and has positive topological entropy. In the concrete example of the modular group Γ = PSL(2,Z) we extend the link of periodic geodesics with knot theory, discovered by Ghys, to the integrable region. In the second part, we consider the integrable generalisations of the Dirac magnetic monopole on sphere S² with general metric and constant non-zero density of magnetic field. We complete the local classification results of such systems by Ferapontov, Sayles and Veselov, extending them to the analytic integrable systems on the topological sphere. In the limiting even case we have the new integrable two-centre problem on the usual round sphere in the external field of Dirac magnetic monopole.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
Keywords: SL(2,R) ; Integrability of geodesic flow ; Quantum integrable systems ; New 2-center problem on sphere ; Euler-Dirac ; Magnetic field