  We formulate the Lagrangian perturbation theory to solve the non-linear
dynamics of self-gravitating fluid within the framework of the post-Newtonian
approximation in general relativity, using the (3+1) formalism. Our formulation
coincides with Newtonian Lagrangian perturbation theory developed by Buchert
for the scale much smaller than the horizon scale, and with the gauge invariant
linearized theory in the longitudinal gauge conditions for the linear regime.
These are achieved by using the gauge invariant quantities at the initial time
when the linearized theory is valid enough. The post-Newtonian corrections in
the solution of the trajectory field of fluid elements are calculated in the
explicit forms. Thus our formulation allows us to investigate the evolution of
the large-scale fluctuations involving relativistic corrections from the early
regime such as the decoupling time of matter and radiation until today. As a
result, we are able to show that naive Newtonian cosmology to the structure
formation will be a good approximation even for the perturbations with scales
not only inside but also beyond the present horizon scale in the longitudinal
coordinates. Although the post-Newtonian corrections are small, it is shown
that they have a growing transverse mode which is not present in Newtonian
theory as well as in the gauge invariant linearized theory. Such post-Newtonian
order effects might produce characteristic appearances of the large-scale
structure formation, for example, through the observation of anisotropies of
the cosmic microwave background radiation (CMB). Furthermore since our approach
has a straightforward Newtonian limit, it will be also convenient for numerical
implementation based on the presently available Newtonian simulation. ......
