Euler-Lagrange Elasticity with Dynamics

The equations of Euler-Lagrange elasticity describe elastic deformations without reference to stress or strain. These equations as previously published are applicable only to quasi-static de-formations. This paper extends these equations to include time dependent deformations. To ac-complish this, an appropriate Lagrangian is defined and an extrema of the integral of this Lagran-gian over the original material volume and time is found. The result is a set of Euler equations for the dynamics of elastic materials without stress or strain, which are appropriate for both finite and infinitesimal deformations of both isotropic and anisotropic materials. Finally, the resulting equations are shown to be no more than Newton's Laws applied to each infinitesimal volume of the material