The Lagrangian formulation of nonlocal mechanics is investigated on the basis of a new nonlocal argument that is defined as a nonlocal residual satisfying the zero mean condition. The nonlocal Euler–Lagrangian equation is derived from the Hamilton’s principle, and some new characters are found. First integral of the nonlocal Euler–Lagrangian equation is determined with the form of energy-momentum tensor. Finally, a quadratic Lagrangian function including the nonlocal residual-based argument is used to qualitatively predict the dispersion of one dimensional nonlocal elastic wave.