  We consider symplectic time integrators in numerical General Relativity and
discuss both free and constrained evolution schemes. For free evolution of
ADM-like equations we propose the use of the Stoermer-Verlet method, a standard
symplectic integrator which here is explicit in the computationally expensive
curvature terms. For the constrained evolution we give a formulation of the
evolution equations that enforces the momentum constraints in a holonomically
constrained Hamiltonian system and turns the Hamilton constraint function from
a weak to a strong invariant of the system. This formulation permits the use of
the constraint-preserving symplectic RATTLE integrator, a constrained version
of the Stoermer-Verlet method.
  The behavior of the methods is illustrated on two effectively 1+1-dimensional
versions of Einstein's equations, that allow to investigate a perturbed
Minkowski problem and the Schwarzschild space-time. We compare symplectic and
non-symplectic integrators for free evolution, showing very different numerical
behavior for nearly-conserved quantities in the perturbed Minkowski problem.
Further we compare free and constrained evolution, demonstrating in our
examples that enforcing the momentum constraints can turn an unstable free
evolution into a stable constrained evolution. This is demonstrated in the
stabilization of a perturbed Minkowski problem with Dirac gauge, and in the
suppression of the propagation of boundary instabilities into the interior of
the domain in Schwarzschild space-time.
