  A possible definition of strong/symmetric hyperbolicity for a second-order
system of evolution equations is that it admits a reduction to first order
which is strongly/symmetric hyperbolic. We investigate the general system that
admits a reduction to first order and give necessary and sufficient criteria
for strong/symmetric hyperbolicity of the reduction in terms of the principal
part of the original second-order system. An alternative definition of strong
hyperbolicity is based on the existence of a complete set of characteristic
variables, and an alternative definition of symmetric hyperbolicity is based on
the existence of a conserved (up to lower order terms) energy. Both these
definitions are made without any explicit reduction. Finally, strong
hyperbolicity can be defined through a pseudo-differential reduction to first
order. We prove that both definitions of symmetric hyperbolicity are equivalent
and that all three definitions of strong hyperbolicity are equivalent (in three
space dimensions). We show how to impose maximally dissipative boundary
conditions on any symmetric hyperbolic second order system. We prove that if
the second-order system is strongly hyperbolic, any closed constraint evolution
system associated with it is also strongly hyperbolic, and that the
characteristic variables of the constraint system are derivatives of a subset
of the characteristic variables of the main system, with the same speeds.
