In the study of Hamiltonian flows (classical mechanics) the motion of a point (particle) is determined by a single function (the energy). In this project we will study a new evolution equation for star-shaped surfaces that is defined in terms of Hamiltonian flows. To each starshaped surface one can associate a simple function which measures how far the surface is from a round sphere. This function defines a Hamiltonian flow which moves the surface, which changes the flow, which moves the surface, .... etc. The resulting evolution of the original surface is governed by a differential equation which in two dimensions equals the standard heat equation, and in higher dimensions resembles it. The hope is that, like solutions of the heat equation, the evolving surface quickly converges to an interesting equilibrium. The first goal of the project will be to model the relevant equation in order to study the convergence of the solutions and the special features of the equilibrium surfaces.
Completion of Calculus 3 and familiarity with differential equations at the level of Math 285. Familiarity with software like Matlab and Python is also desired but not expected.