Josh completed his BSc in Mathematics at Royal Holloway, University of London, before earning an MSc in Mathematics at King’s College London. His research interests can be described broadly as mathematical physics. Josh wrote his MSc dissertation on “The Direct Method in the Calculus of Variations” which examined the direct method via applications to the Thomas-Fermi problem and modelling rotating star systems. He is also currently interested in the Vlasov-Poisson equations which are used to model galactic dynamics.

Outside of maths, Josh enjoys video/tabletop/ trading card games, reading, coffee and going on long nature walks.

Project title:
Towards the Construction of Axisymmetric Steady States of the Vlasov–Poisson System with Three Integrals of Motion

Supervisor(s):
Matthew Schrecker, Miles Wheeler

Project description:
The Vlasov–Poisson system is a pair of coupled partial diferential equations that describe the collective dynamics of stars within a galaxy. Since galactic evolution occurs over extremely long time-scales, it is natural to assume that solutions are time-independent and refer to such solutions as steady states. A classical result, known as Jeans’ theorem, asserts that steady states can be expressed in terms of conserved quantities, or integrals of motion, which remain constant along the orbits of the system. Identifying all of the independent integrals therefore provides a complete classifcation of steady-state solutions.

The case of spherical (radial) symmetry is well understood: steady states can be fully classified using the two conserved quantities total energy, E, and full angular momentum, L. In the axisymmetric setting, full angular momentum conservation is lost, but its z-component, L_z, remains conserved. Some progress has been made in constructing families of axisymmetric steady states depending on these two integrals that are perturbations of known radial solutions. However, numerical simulations suggest that to fully capture the dynamics of the system, a third independent integral—closely related to the full angular momentum—is required.

The central aim of this PhD project is to extend existing results on axisymmetric solutions near radial symmetry by constructing steady states of the Vlasov–Poisson system that depend on three independent integrals of motion.