  The post-Newtonian (PN) approximation, based on the assumptions of weak
gravitational fields and slow motions, provides a way to estimate general
relativistic effects in the fully nonlinear evolution stage of the large-scale
cosmic structures. We extend Chandrasekhar's first order PN (1PN) hydrodynamics
based on the Minkowski background to the Robertson-Walker background. We assume
the presence of Friedmann's cosmological spacetime as a background. In the
background we include the three-space curvature, the cosmological constant and
general pressure. In the Newtonian order and 1PN order we include general
pressure, stress, and flux. The Newtonian hydrodynamic equations appear
naturally in the 0PN order. The spatial gauge degree of freedom is fixed in a
unique manner and the basic equations are arranged without taking the temporal
gauge condition. In this way we can conveniently try alternative temporal gauge
conditions. We investigate a number of temporal gauge conditions under which
all the remaining variables are equivalently gauge-invariant. Our aim is to
present the fully nonlinear 1PN equations in a form suitable for implementation
in conventional Newtonian hydrodynamic simulations with minimal extensions. The
1PN terms can be considered as relativistic corrections added to the well known
Newtonian equations. The proper arrangement of the variables and equations in
combination with suitable gauge conditions would allow the possible future 1PN
cosmological simulations to become more tractable. Our equations and gauges are
arranged for that purpose. We suggest ways of controlling the numerical
accuracy. The typical 1PN order terms are about $10^{-6} \sim 10^{-4}$ times
smaller than the Newtonian terms.
