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In this paper, we derive from the principle of least action the equation ofmotion for a continuous medium with regularized density field in the context ofmeasures. The eventual equation of motion depends on the order in whichregularization and the principle of least action are applied. We obtain twodifferent equations, whose discrete counterparts coincide with the scheme usedtraditionally in the Smoothed Particle Hydrodynamics (SPH) numerical method(e.g. Monaghan), and with the equation treated by Di Lisio et al.,respectively. Additionally, we prove the convergence in the Wassersteindistance of the corresponding measure-valued evolutions, moreover providing theorder of convergence of the SPH method. The convergence holds for a generalclass of force fields, including external and internal conservative forces,friction and non-local interactions. The proof of convergence is illustratednumerically by means of one and two-dimensional examples.

From continuum mechanics to SPH particle systems and back: Systematic derivation and convergence