Discrete Dirac structures and discrete Lagrange--Dirac dynamical systems in mechanics

In this paper, we propose the concept of $(\pm)$-discrete Dirac structures over a manifold, where we define $(\pm)$-discrete two-forms on the manifold and incorporate discrete constraints using $(\pm)$-finite difference maps. Specifically, we develop $(\pm)$-discrete induced Dirac structures as discrete analogues of the induced Dirac structure on the cotangent bundle over a configuration manifold, as described by Yoshimura and Marsden (2006). We demonstrate that $(\pm)$-discrete Lagrange--Dirac systems can be naturally formulated in conjunction with the $(\pm)$-induced Dirac structure on the cotangent bundle. Furthermore, we show that the resulting equations of motion are equivalent to the $(\pm)$-discrete Lagrange--d'Alembert equations proposed in Cort\'es and Mart\'inez (2001) and McLachlan and Perlmutter (2006). We also clarify the variational structures of the discrete Lagrange--Dirac dynamical systems within the framework of the $(\pm)$-discrete Lagrange--d'Alembert--Pontryagin principle. Finally, we validate the proposed discrete Lagrange--Dirac systems with some illustrative examples of nonholonomic systems through numerical tests.