Extension the Noether's theorem to Lagrangian formulation with nonlocality

TL;DR

This paper extends Noether's theorem to a nonlocal Lagrangian framework, deriving conservation laws in nonlocal elasticity and analyzing their localization properties.

Contribution

It introduces a new nonlocal argument in the Lagrangian, extending Noether's theorem to derive conservation laws in nonlocal elasticity.

Findings

Conservation laws are valid as integrals over the entire domain.

Localization of conservation laws depends on the existence of nonlocal residuals.

Not all conservation laws correspond to local equilibrium equations.

Abstract

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.