Phil Holmes - Non-holonomic and piecewise-holonomic mechanical systems

Nonholonomic (velocity dependent) constraints can lead to asymptotically stable motions in certain conservative mechanical systems; the Chaplygin sleigh is a canonical example. In studying models for legged locomotion, piecewise-holonomic constraints (due to intermittent foot placements) are typical. The resulting hybrid dynamical systems include flows along a smooth vectorfield punctuated by impulsive jumps governed by discrete `collision maps.' They may be viewed as generalisations of billiards-type problems. Such systems can also exhibit partial asymptotic stability, even while conserving total energy. I will describe joint work with Michael Coleman (Cornell University) and John Schmitt (Princeton University) on a discrete sister to the Chaplygin sleigh, and on a simple model for rapidly running insects, which illustrate this phenomenon.