We present a novel application of the extended Fast Action Minimization method (eFAM) aimed at assessing the role of the environment in shaping galaxy evolution. We validate our approach by testing eFAM predictions against the Magneticum hydrodynamical simulation. We consider the z~0 snapshot of the simulation as our observed catalogue and use the reconstructed trajectories of galaxies to model the evolution of cosmic structures. At the statistical level, the fraction of volume (VFF) occupied by voids, sheets, filaments, and clusters in the reconstructed catalogues agrees within $1\sigma$ with the VFF estimated from the high-redshift snapshots of the simulation. The local accuracy of eFAM structures is evaluated by computing their purity with respect to the simulated catalogues, P, at the cells of a regular grid. Up to z=1.2, clusters have 0.58