Master of Science (MSc)
Mathematics for Science and Finance
Faculty of Science
The motion of objects in the sky has captured the attention of scientists and mathematicians since classical times. The problem of determining their motion has been dubbed the Kepler problem, and has since been generalized into an abstract problem of dynamical systems. In particular, the question of whether a classical system produces closed and bounded orbits is of importance even to modern mathematical physics, since these systems can often be analysed by hand. The aforementioned question was originally studied by Bertrand in the context of celestial mechanics, and is therefore referred to as the Bertrand problem. We investigate the qualitative behaviour of solutions to the generalized Kepler problem, in which particles travel in an abstract space called a manifold. We find that although Bertrand's results do not generalize to even the most simple non-trivial example of a complex manifold, we can partially reduce the problem through the use of a function called a momentum map. We then study the generalized Kepler problem on pseudo-Riemannian surfaces of revolution. In this case we are able to demonstrate that a generalization of Bertrand's theorem holds, and we compute explicit expressions for the shape and period of the orbits. Furthermore, we compute a generalization of the Laplace-Runge-Lenz vector, which allows one to determine all solutions to the equations of motion in terms of three constant quantities.
Astwood, Michael R., "The Kepler Problem on Complex and Pseudo-Riemannian Manifolds" (2022). Theses and Dissertations (Comprehensive). 2503.
Dynamical Systems Commons, Dynamic Systems Commons, Geometry and Topology Commons, Ordinary Differential Equations and Applied Dynamics Commons, Other Astrophysics and Astronomy Commons, Other Physics Commons