We consider the minimum time control of dynamical systems with slow and fast state variables. With applications to perturbations of integrable systems in mind, we focus on the case of problems with one or more fast angles, together with a small drift on the slow part modelling a so-called secular evolution of the slow variables. According to Pontrjagin maximum principle, minimizing trajectories are projections on the state space of Hamiltonian curves. In the case of a single fast angle, it turns out that, provided 9 the drift on the slow part of the original system is small enough, time minimizing trajectories can be approximated by geodesics of a suitable metric. 11 As an application to space mechanics, the effect of the J2 term in the Earth potential on the control of a spacecraft is considered. In ongoing work, we 13 also address the more involved question of systems having two fast angles.