Cosmological dynamics: from the Eulerian to the Lagrangian frame - I.
  Newtonian approximation
  We analyse the non-linear gravitational dynamics of a pressure-less fluid in
the Newtonian limit of General Relativity in both the Eulerian and Lagrangian
pictures. Starting from the Newtonian metric in the Poisson gauge, we transform
to the synchronous and comoving gauge and obtain the Lagrangian metric within
the Newtonian approximation. Our approach is fully non-perturbative, which
implies that if our quantities are expanded according to the rules of standard
perturbation theory, all terms are exactly recovered at any order in
perturbation theory, only provided they are Newtonian. We explicitly show this
result up to second order and in both gauges. Our transformation clarifies the
meaning of the change of spatial and time coordinates from the Eulerian to the
Lagrangian frame in the Newtonian approximation.
