On the Infinite Particle Limit in Lagrangian Dynamics and Convergence of Optimal Transportation Meshfree Methods

Abstract

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.

Galerkin local maximum entropy method

“…For rigorous convergence proofs, see also the work of Schmidt. 40 For application of the LME approximation scheme in an SPH setting, see the works of Hartmann et al 41,42 This contribution builds up onto the master's thesis by Dreier. 43…”

“…The method further ensures proper conservation laws by using an optimal transportation formulation. For rigorous convergence proofs, see also the work of Schmidt …”

Discontinuous variational time integrators for complex multibody collisions

“…Q, the precise meaning of the symbol to be disambiguated by the context. || Confirm for example, [5,17,19] for functional analytical frameworks for Lagrangian mechanics. ** Here and subsequently, we proceed formally and assume that invoked mathematical operations are well defined.…”

“…In this section, we summarize the formulation of constrained Lagrangian mechanics that provides the basis for subsequent developments (cf., e.g., for additional background). We particularly wish to account for collisions involving multibody contact between possibly nonsmooth bodies.…”