  The ADM Hamiltonian formulation of general relativity with prescribed lapse
and shift is a weakly hyperbolic system of partial differential equations. In
general weakly hyperbolic systems are not mathematically well posed. For well
posedness, the theory should be reformulated so that the complete system,
evolution equations plus gauge conditions, is (at least) strongly hyperbolic.
Traditionally, reformulation has been carried out at the level of equations of
motion. This typically destroys the variational and Hamiltonian structures of
the theory. Here I show that one can extend the ADM formalism to (i)
incorporate the gauge conditions as dynamical equations and (ii) affect the
hyperbolicity of the complete system, all while maintaining a Hamiltonian
description. The extended ADM formulation is used to obtain a strongly
hyperbolic Hamiltonian description of Einstein's theory that is generally
covariant under spatial diffeomorphisms and time reparametrizations, and has
physical characteristics. The extended Hamiltonian formulation with 1+log
slicing and gamma--driver shift conditions is weakly hyperbolic.
