Multiple Instantaneous Collisions in a Variational Framework

Variational integrators have become of interest to the controls community, particularly since in recent years they have been used in the context of optimal control. This paper discusses how to resolve simultaneous impacts in a variational context, assuming that all boundaries of the impacting bodies are differentiable. We demonstrate how this analysis works for arbitrary numbers of simultaneous impacts and derive discrete time algorithms for resolving these impacts numerically. We illustrate our results, both in the continuous and discrete domain, using Newton's cradle as an example.