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Consider a partition of $R^n$ into finitely many polyhedral regions $D_i$ andassociated drift vectors $\mu_i\in R^n$. We study ``hybrid'' dynamical systemswhose trajectories have a constant drift, $\dot x=\mu_i$, whenever $x$ is inthe interior of the $i$th region $D_i$, and behave consistently on the boundarybetween different regions. Our main result asserts that if such a system isnonexpansive (i.e., if the Euclidean distance between any pair of trajectoriesis a nonincreasing function of time), then the system must be conservative,i.e., its trajectories are the same as the trajectories of the negativesubgradient flow associated with a potential function. Furthermore, thispotential function is necessarily convex, and is linear on each of the regions$D_i$. We actually establish a more general version of this result, by makingseemingly weaker assumptions on the dynamical system of interest.

Nonexpansive Piecewise Constant Hybrid Systems are Conservative