\(\Gamma \)-convergence of variational integrators for constrained systems (Q839442)

Mechanical systems in an \(n\)-dimensional configuration space that can be described by the evolution \(t\mapsto u(t)\in \mathbb{R}^n\), with time \(t\) subject to the potential \(V: \mathbb{R}^n\to \mathbb{R}\) and subject to holonomic constraints are investigated. The Lagrangian mechanics, the physical trajectories of this motion arise as stationary points of the corresponding action functional \(I\) which is the kinetic energy minus the potential energy integrated along the trajectory. In the presence of holonomic constraints, modeled by the so-called constraint manifold \(M\subset\mathbb{R}^n\), requiring in addition that the trajectories must lie on \(M\), gives rise to the constrained functional \(I_M\). For a physical system described by a motion in an energy landscape under holonomic constraints, the \(\Gamma\)-convergence of variational integrators to the corresponding continuum action functional and the convergence properties of solutions of the discrete Euler-Lagrange equations for stationary points of the continuum problem are studied. The convergence result is illustrated with examples of mass point systems and flexible multibody dynamics.