  Well-posedness of the initial (boundary) value problem is an essential
property, both of meaningful physical models and of numerical applications. To
prove well-posedness of wave-type equations their level of hyperbolicity is an
essential ingredient. We develop helpful tools and classify a large class of
Hamiltonian versions of Einstein's equations with live gauge conditions with
respect to their hyperbolicity. Finally we find a symmetric hyperbolic
Hamiltonian formulation that allows for gauge conditions which are similar to
the puncture gauge.
