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We investigate the Lagrangian perturbation theory of a homogeneous andisotropic universe in the non-relativistic limit, and derive the solutions upto the fourth order. These solutions are needed for example for thenext-to-leading order correction of the (resummed) Lagrangian matterbispectrum, which we study in an accompanying paper. We focus on flatcosmologies with a vanishing cosmological constant, and provide an in-depthdescription of two complementary approaches used in the current literature.Both approaches are solved with two different sets of initial conditions---bothappropriate for modelling the large-scale structure. Afterwards we consideronly the fastest growing mode solution, which is not affected by either ofthese choices of initial conditions. Under the reasonable approximation thatthe linear density contrast is evaluated at the initial Lagrangian position ofthe fluid particle, we obtain the nth-order displacement field in the so-calledinitial position limit: the nth order displacement field consists of 3(n-1)integrals over n linear density contrasts, and obeys self-similarity. Then, wefind exact relations between the series in Lagrangian and Eulerian perturbationtheory, leading to identical predictions for the density contrast and thepeculiar-velocity divergence up to the fourth order.

Lagrangian perturbations and the matter bispectrum I: fourth-order model for non-linear clustering