In this paper, we propose the concept of (±)-discrete Dirac structures over a manifold by introducing (±)-discrete two-forms and incorporate discrete constraints via (±)-finite difference approximations. In particular, we develop (±)-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 J. Geom. Phys. 57, 133–156 (2006). We demonstrate that (±)-discrete Lagrange–Dirac systems can be naturally formulated in conjunction with these (±)-induced Dirac structure. Furthermore, we show that the resulting equations of motion are equivalent to the discrete Lagrange–d’Alembert equations proposed in Cortés and Martínez Nonlinearity 14, 1365–1392 (2001) and McLachlan and Perlmutter J. Nonlinear Sci. 16, 283–328 (2006). We also clarify the variational structure of the discrete Lagrange–Dirac dynamical systems within the framework of the (±)-discrete Lagrange–d’Alembert–Pontryagin principle. Finally, we validate the proposed discrete Lagrange–Dirac systems through numerical tests involving illustrative examples of nonholonomic systems.
Peng et al. (Mon,) studied this question.