  In this paper we derive the dynamic equations of a race-car model via
Lie-group methods. Lie-group methods are nowadays quite familiar to
computational dynamicists and roboticists, but their diffusion within the
vehicle dynamics community is still limited. We try to bridge this gap by
showing that this framework merges gracefully with the Articulated Body
Algorithm (ABA) and enables a fresh and systematic formulation of the vehicle
dynamics. A significant contribution is represented by a rigorous
reconciliation of the ABA steps with the salient features of vehicle dynamics,
such as road-tire interactions, aerodynamic forces and load transfers. The
proposed approach lends itself both to the definition of direct simulation
models and to the systematic assembly of vehicle dynamics equations required,
in the form of equality constraints, in numerical optimal control problems. We
put our approach on a test in the latter context which involves the solution of
minimum lap-time problem (MLTP). More specifically, a MLTP for a race car on
the N\"urburgring circuit is systematically set up with our approach. The
equations are then discretized with the direct collocation method and solved
within the CasADi optimization suite. Both the quality of the solution and the
computational efficiency demonstrate the validity of the presented approach.
