Numerische Mathematik manuscript No. (will be inserted by the editor) A. M. Bloch · I. I. Hussein · M. Leok · A. K. Sanyal A Discrete Variational Integrator for Optimal Control Problems on SO(3) Received: date / Revised: date Abstract In this paper we study a discrete variational optimal control prob- lem for the rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange–d’Alembert prin- ciple, a process that better approximates the equations of motion. Within the discrete-time setting, these two approaches are not equivalent in general. The kinematics are discretized using a natural Lie-algebraic formulation that guarantees that the flow remains on the Lie group SO(3) and its algebra so(3). We use Lagrange’s method for constrained problems in the calculus of vari- ations to derive the discrete-time necessary conditions. We give a numerical example for a three-dimensional rigid body maneuver. A. Bloch Alexander Ziwet Collegiate Professor of Mathematics and Department Chair Department of Mathematics University of Michigan E-mail: abloch@umich.edu I. Hussein Assistant Professor Worcester Polytechnic Institute E-mail: ihussein@uiuc.edu M. Leok Hildebrandt Research Assistant Professor Department of Mathematics University of Michigan E-mail: mleok@umich.edu A. Sanyal Postdoctoral Research Associate Mechanical and Aerospace Engineering Arizona State University E-mail: sanyal@asu.edu