  This thesis details an effort to generate astrophysically interesting
solutions to the two-body problem in General Relativity. The thesis consists of
two main parts. The first part presents an analytical variational principle for
describing binary neutron stars undergoing irrotational fluid flow. The
variational principle is a powerful tool for generating accurate estimates of
orbital parameters for neutron stars in quasi-equilibrium orbits. The second
part of the thesis details the numerical application of the variational
principle by solving the initial value problem for binary black holes in quasi-
equilibrium circular orbits. The analysis draws from the novel ``puncture''
method of describing the black holes, and relies on nonlinear adaptive
multigrid techniques for generating numerical results. We arrive at two
important conclusions. First, the analytical variational principle describing
binary neutron stars in irrotational motion provides a road map for future
numerical simulations, and also lends credence to previous simulations by other
authors. Second, the numerical application and description of binary black
holes in quasi-equilibrium circular orbits simplifies the analyses of previous
authors, and allows for the imposition of realistic boundary data in
simulations with relatively high grid densities. Both the variational principle
and its application are used to generate accurate estimates of the orbital
parameters.
