  We present a numerical framework for modeling extended hyperelastic bodies
based on a Lagrangian formulation of general relativistic elasticity theory. We
use finite element methods to discretize the body, then use the semi--discrete
action to derive ordinary differential equations of motion for the discrete
nodes. The nodes are evolved in time using fourth--order Runge--Kutta. We
validate our code against the normal modes of oscillation of a hyperelastic
sphere, which are known analytically in the limit of small (linear), slow
(Newtonian) oscillations. The algorithm displays second order convergence. This
numerical framework can be used to obtain the orbital motion and internal
dynamics of a hyperelastic body of any shape, for any spacetime metric, and for
varying hyperelastic energy models.
