  We present a new formulation to construct numerically equilibrium
configurations of rotating stars in general relativity. Having in mind the
application to their quasi static evolutions, we adopt a Lagrangian formulation
of our own devising, in which we solve force balance equations to seek for the
positions of fluid elements assigned to the grid points, instead of the
ordinary Eulerian formulation. Unlike previous works in the literature, we do
not employ the first integral of the Euler equation, which is not obtained by
an analytic integration in general. We assign a mass, specific angular momentum
and entropy to each fluid element in contrast to the previous methods, in which
the spatial distribution of the angular velocity or angular momentum is
specified. Those distributions are determined after the positions of all fluid
elements (or grid points) are derived in our formulation. We solve the large
system of algebraic nonlinear equations that are obtained by discretizing the
time-independent Euler and Einstein equations in the finite-elements method by
using our new multi-dimensional root-finding scheme, named the W4 method. To
demonstrate the capability of our new formulation, we construct some rotational
configurations both barotropic and baroclinic. We also solve three evolutionary
sequences that mimic the cooling, mass-loss, and mass-accretion as simple toy
models.
