Abstract: This paper proposes a simplified $<$i$>$position-based$<$/i$>$ physics that allows us to rapidly generate "piles" or "clumps" of many objects: local energy minima under a variety of potential energy functions. We can also generate plausible motions for many highly interacting objects from arbitrary starting positions to a local energy minimum. We present an efficient and numerically stable algorithm for carrying out position-based physics on spheres and non-rotating polyhedra through the use of linear programming. This algorithm is a generalization of an algorithm for finding tight packings of (non-rotating) polygons in two dimensions. This work introduces linear programming as a useful tool for graphics animation. As its name implies, position-based physics does not contain a notion of velocity, and thus it is not suitable for simulating the motion of free-flying, unencumbered objects. However, it generates realistic motions of "crowded" sets of objects in confined spaces, and it does so at least two orders of magnitude faster than other techniques for simulating the physical motions of objects. Even for unconfined objects, the new algorithm can rapidly generate realistic "piles" and "clumps."