Nonsmooth Lagrangian Systems
This chapter is dedicated to Lagrangian dynamical systems subject to bilateral and unilateral constraints, which we may name Complementarity Lagrangian Systems. It starts with the analysis of the contact problem, for perfect bilateral and unilateral constraints. Then Moreau’s sweeping process is introduced in detail. Coulomb’s model of friction is described, and complementarity problems and systems are presented with examples from Mechanics, Circuits and Optimal Control. The chapter continues with the analysis of the contact problem when friction acts on the system, with a particular emphasis on the so-called Painlevé paradoxes. An introduction to various existing methods for the numerical integration of nonsmooth systems with complementarity relations ends the chapter.