Euler equations as consequence of Newton's laws

Classical mechanics, as seen by mathematics, is most developed in two extreme cases – mechanics of point particles and continuum mechanics. We give two rigorous mathematical examples of direct connection between these two theories, leaving aside stochastic dynamics, thermodynamics, kinetic equations and BBGKY hierarchy. Namely, we consider system of N point particles with the interaction of Lennard-Jones type. In the large N limit we deduce the system of three Euler equations for the resulting “string”. It is rather surprising, that the resulting explicit expression of pressure in terms of the density corresponds to one-dimensional Chaplygin gas, which is actively discussed in the physical literature in connection with the string theory and with models of dark matter and dark energy. Also we will demonstrate video of the string dynamics obtained by numerical modelling.