I'm doing a research project for a class in advanced differential geometry, and have chosen to cover dynamical systems. However, I've found that the relationship between dynamical systems is limited to exploring flows on manifolds in the most abstract sense. As a result, I was hoping on expand on my topic by including general relativity (since it seems to be strongly based on differential geometry). Can anybody think of any dynamical systems in General Relativity that I can play around with? Even suggestions on how to formulate dynamical systems out of problems that have been solved in a purely physical sense would be helpful. For example, I'm working on trying to analyze the dynamics of a massive particle outside of a blackhole using the Schwarzschild metric. I know how to do this physically, but am not entirely sure as how to form the corresponding dynamical system. Any ideas?