In this paper, we employ measure theory to derive from the principle of least action the equation of motion for a continuum with regularized density field. The eventual equation of motion depends on the order in which regularization and the principle of least action are applied. We obtain two different equations, whose discrete counterparts coincide with the scheme used traditionally in the Smoothed Particle Hydrodynamics (SPH) numerical method, and with the equation treated by Di Lisio et al. in 1998, respectively. Additionally, we prove the convergence in the Wasserstein distance of the corresponding measure-valued evolutions, moreover providing the order of convergence of the SPH method. The convergence holds for a general class of force fields, including external and internal conservative forces, friction and non-local interactions. The proof of convergence is illustrated numerically by means of one and two-dimensional examples.
Keywords: Smoothed Particle Hydrodynamics, principle of least action, Wasserstein distance, measure-valued equations, convergence rate