Extension the Noether's theorem to Lagrangian formulation with nonlocality

A Lagrangian formulation with nonlocality is investigated in this paper. The nonlocality of the Lagrangian is introduced by 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. The Noether's theorem is extended to this Lagrangian formulation with nonlocality. With the help of the extended Noether's theorem, the conservation laws relevant to energy, linear momentum, angular momentum and the Eshelby tensor are determined in the nonlocal elasticity associated with the mechanically based constitutive model. The results show that the conservation laws exist only in the form of the integral over the whole domain occupied by body. The localization of the conservation laws is discussed in detail. We demonstrate that not every conservation law corresponds to a local equilibrium equation. Only when the nonlocal residual of conservation current exists, can a conservation law be transformed into a local equilibrium equation by localization.