Symmetry-preserving discretization of variational field theories

The present paper develops a variational theory of discrete fields defined on abstract cellular complexes. The discrete formulation is derived solely from a variational principle associated to a discrete Lagrangian density on a discrete bundle, and developed up to the notion of symmetries and conservation laws for solutions of the discrete field equations. The notion of variational integrator for a Cauchy problem associated to this variational principle is also studied. The theory is then connected with the classical (smooth) formulation of variational field theories, describing a functorial method to derive a discrete Lagrangian density from a smooth Lagrangian density on a Riemannian fibered manifold, so that all symmetries of the Lagrangian turn into symmetries of the corresponding discrete Lagrangian. Elements of the discrete and smooth theories are compared and all sources of error between them are identified. Finally the whole theory is illustrated with the discretization of the classical variational formulation of the kinematics of a Cosserat rod.