Distributed manipulation systems induce motions on objects through the application of many external forces. An actuator array performs distributed manipulation using a planar array of many stationary elements, which cooperate to manipulate larger objects through the generation of a programmable force field. This paper defines and studies a class of manipulation force fields called quadratic potential fields. Methodology is developed to decompose these fields into simpler components and superimpose them both vectorially and geometrically, simplifying the design of manipulation strategies. Application to trajectory following is also presented.