  With the aim of deriving symmetric hyperbolic free-evolution systems for GR
that possess Hamiltonian structure and allow for the popular puncture 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 gauge
conditions which are very similar to the puncture gauge.
