This paper deals with an efficient numerical method for the fully lubricated line contact between a rotating, deformable cylinder and a rigid surface. By exploiting the dynamic variation structure of this non-linear problem the deformation of and the pressure at the free, contact boundary are calculated. The dynamic formulation leads in a natural way to an iterative procedure, where the evolution from one iterate to a subsequent one is governed by a minimization problem. Physically, the Euler-Lagrange equation expresses the fact that the mass has to be conserved. For this reason, in contrast with earlier approaches, mass flux defects do not occur here. The proposed dynamic algorithm starts with the calculation of the lubricated contact between a rigid cylinder and the rigid surface. Then the stiffness of the cylinder is lessened until the desired value is reached, where after the loading on the cylinder is increased by moving it towards the rigid surface. The effort to proceed in time is significantly reduced by preconditioning: the discretized Euler-Lagrange equation is multiplied by an approximation of the inverse of the global operator governing the deflection of the cylinder. In this way, solutions that are comparable to large-time (or super-) computer computations can be calculated on a PC.