Atom and nanoparticle arrays trapped in optical lattices are shown to be capable of sustaining collective oscillations of frequency proportional to the strength of the external light field. The spectrum of these oscillations determines the mechanical stability of the arrays. This phenomenon is studied for dimers, strings, and two-dimensional planar arrays. Laterally confined particles free to move along an optical channel are also considered as an example of collective motion in

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... in partially-confined systems. The fundamental concepts of dynamical response in optical matter introduced here constitute the basis for potential applications to quantum information technology and signal processing. Experimental realizations of these systems are proposed.