Discrete Routh Reduction and Discrete Exterior Calculus

This paper will review recent advances in the formulation of a discrete version of geometric mechanics, based on the discretization of Hamilton’s variational principle, and progress that has been made in reduction theory for discrete variational mechanics in the form of Discrete Routh Reduction. To place discrete geometric mechanics on a firm mathematical foundation, we propose to develop a discrete exterior calculus for simplicially triangulated compact orientable spaces. This will involve the isomorphism between simplicial cohomology and de Rham cohomology via the Whitney and de Rham maps, as well as additional techniques from simplicial algebraic topology, and groupoid theory. Practical applications of a discrete exterior calculus include natural discretizations of the vector operators, div, grad, and curl. These discrete operators will automatically satisfy the standard vector identities, since the coboundary operator from which they are derived is nilpotent of second order. In parallel, extensions to the variational mechanics framework to encompass multiscale and spatio-temporally adaptive techniques will be explored to make variational integrators more competitive for practical applications.