  We consider Lagrangian systems in the limit of infinitely many particles. It
is shown that the corresponding discrete action functionals Gamma-converge to a
continuum action functional acting on probability measures of particle
trajectories. Also the convergence of stationary points of the action is
established. Minimizers of the limiting functional and, more generally,
limiting distributions of stationary points are investigated and shown to be
concentrated on orbits of the Euler-Lagrange flow. We also consider time
discretized systems. These results in particular provide a convergence analysis
for optimal transportation meshfree methods for the approximation of particle
flows by finite discrete Lagrangian dynamics.
