Explicit Momentum-conserving Integrator for Dynamics of Rigid Bodies Approximating the Midpoint Lie Algorithm P. Krysl ∗ May 4, 2004 1 Introduction The focus of this work is the initial value problem of the rotational rigid body dy- namics. Often the calculation of forcing (evaluation of the external torques) is com- putationally intensive. For instance, in many molecular dynamics simulations, the evaluation of the forcing may take as much as 90% or more of the CPU time. Con- sequently, methods that limit the number of evaluations of the external torque are of considerable interest, and we are thus led to contemplate algorithms that are explicit in the torque calculation, i.e. the torque is evaluated just once per time step. The main result of this work follows from a reformulation of the midpoint Lie algorithm, which is implicit in the torque calculation as the impulse needs to be evaluated at the unknown midpoint of the incremental rotation. In order to make the algorithm explicit in the torque calculation, we approximate the impulse delivered over the time step with discrete impulses delivered at either the beginning of the time step or at the end of the time step. Therefore, we obtain two related variants, both of which are explicit in the torque calculation, but only first-order in the time step. Both of these algorithms are momentum-conserving and both are symplectic. Therefore, drawing on the properties of the composition of maps, we introduce another algorithm as a composition of these two variants. The resulting algorithm is then explicit, momentum-conserving, symplectic, and second order. Its accuracy is outstanding and consistently matches or exceeds currently known implicit and explicit integrators, as we show on a number of examples. * University of California, San Diego, 9500 Gilman Dr, La Jolla, CA 92093-0085, pkrysl@ucsd.edu 1