## Nonsmooth Lagrangian mechanics and variational collision integrators

R. C. Fetecau, J. E. Marsden, M. Ortiz, and M. West

*SIAM Journal on Applied Dynamical Systems* **2**(3), 381-416, 2003.

Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem. Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the long-time stable energy behavior typical of variational methods is demonstrated.

DOI: 10.1137/S1111111102406038

Full text: FeMaOrWe2003.pdf