Lagrangian formulation of continuum with internal long-range interactions

Based on a new definition of nonlocal variable, this paper establishes the Lagrangian formulation for continuum with internal long-range interactions. Distinguished from the existing theories, the nonlocal term in the Lagrangian formulation automatically satisfies the zero mean condition determined by the action and reaction law. By this formulation, elastic wave in a rod with the internal long-range interactions is investigated. The dispersion of the elastic wave is predicted.