Lagrange-d’Alembert SPARK Integrators for Nonholonomic Lagrangian Systems

Lagrangian systems with ideal nonholonomic constraints can be expressed as implicit index 2 differential-algebraic equations (DAEs) and can be derived from the Lagrange-d’Alembert principle. We define a new nonholonomically constrained discrete Lagrange-d’Alembert principle based on a discrete Lagrange-d’Alembert principle for forced Lagrangian systems. Nonholonomic constraints are considered as first integrals of the underlying forced Lagrangian system of ordinary differential equations. A large class of specialized partitioned additive Runge-Kutta (SPARK) methods for index 2 DAEs satisfies the new discrete principle. Symmetric Lagrange-d’Alembert SPARK integrators of any order can be obtained based for example on Gauss and Lobatto coefficients as already proposed for more general index 2 DAEs.