The consequences of coupling magnetic and elastic degrees of freedom, where spins and deformations are carried by point-like objects subject to local interactions, are studied, theoretically and by detailed numerical simulations. From the constrained Lagrangians we derive consistent equations of motion for the coupled dynamical variables. In order to probe the dynamics of such a system, we consider external perturbations, such as spin transfer torques for the magnetic part, and homogeneous stresses for the elastic part, associated to their corresponding damping. This approach is applied to the study of ultrafast switching processes in anti-ferromagnetic systems, which have recently attracted attention as candidates for anti-ferromagnetic spintronic devices. Our strategy is then checked in simple, but instructive, situations. We carried out numerical experiments to study, in particular, how the magnetostrictive coupling and external stresses affect the nature of the switching processes in a prototype anti-ferromagnetic material.
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