Self-funded PhD opportunities

Geodesics on surfaces and manifolds: caustics and integrability

  • Application end date: Saturday 1 July 2017
  • Funding Availability: Self-funded PhD students only
  • Department: Mathematics
  • PhD Supervisor: Thomas Waters and Andrew Osbaldestin

If we picture a surface such as a sphere or ellipsoid, then in order to understand the geometry of the surface we need some notion of straight lines. Geodesics are a special class of curves in surfaces which play the role of straight lines. They are defined by taking the properties of straight lines and generalising them. One option is to imagine a particle moving across the surface free from any forces; its path would be a geodesic. It is for this reason that geodesics play such a central role in, for example, General Relativity.

Now suppose we choose a point in the surface and consider a spray of geodesics emanating from that point. Due to the curvature of the surface these geodesics may focus along a curve, much like light reflecting off a cup of coffee will focus along a curve on the surface of the liquid. The focussing line is known as a caustic, and the caustic of geodesics is called the conjugate locus.

The equations which describe the geodesics of a surface can be surprisingly complicated, indeed there are simple surfaces for which the geodesic equations cannot be solved. The question of whether we can or cannot solve the equations is known as integrability, and one consequence of non-integrability is the chaotic behaviour of geodesic curves.

This project will make a detailed study of the caustics of geodesics on certain classes of surfaces, a topic of which surprisingly little is known and much remains to be discovered. We will also consider the integrability of the geodesic equations on certain surfaces and seek new surfaces which are integrable. Finally we will combine these two notions and examine what effect non-integrability has on the caustic of geodesics.

The background theory of this project is Differential/Riemannian Geometry on one hand and Lagrangian/Hamiltonian dynamics on the other. We will also use techniques from bifurcation/singularity theory and perturbation methods, and at every stage we will use software to visualise and numerically experiment with this rich and complex topic. The techniques and methods of this project have found application in various fields such as seismology, optimal control theory and crystallographic optics.

How to apply

To apply or make an enquiry, please visit postgraduate research: Mathematics and Physics

Applications should use our standard application forms and follow the instructions given under the ‘Research Degrees’ heading on the following webpage: http://www.port.ac.uk/application-fees-and-funding/applying-postgraduate/#rd

When applying please note the project code: MPHY3350217

Funding Notes: Home/EU applicants only. Please use the online application form and state the project code and studentship title in the personal statement section.

An appropriate first or upper second class honours degree of any United Kingdom university or a recognised equivalent non-UK degree of the same standard honours degree or equivalent in a relevant subject or a master’s degree in an appropriate subject. Exceptionally, equivalent professional experience and/or qualifications will be considered.

References to recent published articles:

1. T. Waters, Bifurcations of the conjugate locus, Journal of Geometry and Physics, submitted.

2. T. Waters, Regular and irregular geodesics on spherical harmonic surfaces, Physica D: Nonlinear Phenomena, 241(5):543-552, 2012.

3. A.Osbaldestin, A.Burbanks and D.Hennig, Transient-chaos induced directed transport in a spatially-open Hamiltonian system, J. Phys. A: Math. And Theoretical, 43(34):345101, 2010.