  We study the relativistic dynamics of a pressure-less and irrotational fluid
of dark matter (CDM) with a cosmological constant ($\Lambda$), up to second
order in cosmological perturbation theory. In our analysis we also account for
vector and tensor perturbations and include primordial non-Gaussianity. We
consider three gauges: the synchronous-comoving gauge, the Poisson gauge and
the total matter gauge, where the first is the unique relativistic Lagrangian
frame of reference, and the latters are convenient gauge choices for Eulerian
frames. Our starting point is the metric and fluid variables in the Poisson
gauge up to second order. We then perform the gauge transformations to the
synchronous-comoving gauge and subsequently to the total matter gauge. Our
expressions for the metrics, densities, velocities, and the gauge generators
are novel and coincide with known results in the limit of a vanishing
cosmological constant.
