Elastodynamics of Multilattices: Field Equations of the Linear Theory as a First Order System

For a one dimensional 2-lattice we write down the momentum equation and the equation ruling the shift vector for the dynamic case as a first order system and characterize it in terms of its hyperbolicity. We use similar arguments for problems of increasing difficulty and tackle: one dimensional 3-lattices, three dimensional 2-lattices, three dimensional 3-lattices and finally the most general case of three dimensional $(n+1)$ $(n+1)$ -lattices. Our approach is confined to the geometrically and materially linear elastodynamic theory and is valid for generic anisotropic materials that have the multilattice structure. The main finding is that, with the assumptions adopted, the presence of each shift vector is related with zero eigenvalues when the system is written as a first order system.