  We formulate a perturbative approximation to gravitational instability, based
on Lagrangian hydrodynamics in Newtonian cosmology. We take account of
`pressure' effect of fluid, which is kinematically caused by velocity
dispersion, to aim hydrodynamical description beyond shell crossing. Master
equations in the Lagrangian description are derived and solved perturbatively
up to second order. Then, as an illustration, power spectra of density
fluctuations are computed in a one-dimensional model from the Lagrangian
approximations and Eulerian linear perturbation theory for comparison. We find
that the results by the Lagrangian approximations are different from those by
the Eulerian one in weakly non-linear regime at the scales smaller than the
Jeans length. We also show the validity of the perturbative Lagrangian
approximations by consulting difference between the first-order and the
second-order approximations.
