Hyperbolic formulations of general relativity with Hamiltonian structure

With the aim of deriving symmetric hyperbolic free-evolution systems for general relativity (GR) that possess Hamiltonian structure and allow for the popular puncture coordinate gauge condition, we analyze the hyperbolicity of Hamiltonian systems. We develop helpful tools which are applicable to either the first order in time, second order in space or the fully second order form of the equations of motion. For toy models we find that the Hamiltonian structure can simplify the proof of symmetric hyperbolicity. In GR we use a special structure of the principal part to prove symmetric hyperbolicity of a formulation that includes conditions which are very similar to the puncture coordinate gauge.