"Lagrangian/Eulerian Description of a Dynamic System"

This paper briefly reveiws the two complementary descriptions of a dynamical system in its phase space as follows: (1) the Lagrangian point-of-view (leading to either a Monte Carlo simulation or a Gibbs set evolution study); and (2) the Eulerian point-of-view (leading to the global analytical equations governing the dynamics of mechanical systems, like the Liouville and the Fokker-Planck equations, (FPE) and to the cell method in a numerical context). It points out the characteristics that a numerical method must show to obtain a correct description of the system dynamics and moves with continuity from deterministic problems to chaotic and/or stochastic situations. The aim of this paper is the implementation of numerical techniques making the study of realistic problems possible. For this purpose, a preliminary academic example deals with the Duffing oscillator to assess the effectiveness of the developed numerical scheme. The second example pursues the assessment of the failure pro...