We study weakly optimal solutions of infinite-horizon variational problems with first-order nonconvex integrands. This is a weakened version of the overtaking optimality criterion. These optimal solutions are closely related to the minimal solutions studied by Moser, Aubry, and Mather. Such solutions have definite rotation number, and we study the relation between the rotation number and the minimal energy growth rate. We establish the existence of a weakly optimal solution for every prescribed initial condition. We also consider discrete-time infinite-horizon periodic control problems in R(n). Analogous to rotation numbers we consider rotation vectors and study minimization of energy growth rate for a prescribed rotation vector. This constrained problem is related to an unconstrained minimization problem that has the same class of minimizers.