Abstract <p>We investigated the application of the full Lagrangian approach to fluid dynamics problems characterised by discontinuities in the acceleration and velocity fields. These discontinuities are typical of processes including jumps in the carrier phase and collisions with solid bodies. We have derived a set of analytical relations that describe the parameters of a cold gas across arbitrary discontinuities of the acceleration and velocity field. The efficiency of the proposed methodology is demonstrated through its application to two specific problems: modelling the distribution of interstellar dust in the heliosphere and analysing collision dynamics with a solid cylindrical surface.</p>