  We extensively develop a perturbation theory for nonlinear cosmological
dynamics, based on the Lagrangian description of hydrodynamics. We solve
hydrodynamic equations for a self-gravitating fluid with pressure, given by a
polytropic equation of state, using a perturbation method up to second order.
This perturbative approach is an extension of the usual Lagrangian perturbation
theory for a pressureless fluid, in view of inclusion of the pressure effect,
which should be taken into account on the occurrence of velocity dispersion. We
obtain the first-order solutions in generic background universes and the
second-order solutions in wider range of a polytropic index, whereas our
previous work gives the first-order solutions only in the Einstein-de Sitter
background and the second-order solutions for the polytropic index 4/3. Using
the perturbation solutions, we present illustrative examples of our formulation
in one- and two-dimensional systems, and discuss how the evolution of
inhomogeneities changes for the variation of the polytropic index.
