  This paper is concerned with a class of time-optimal control problems for the
swing and the ski. We first consider the motion of a man standing on a swing.
For simplicity, we neglect friction and air resistance and assume that the mass
of the swinger is concentrated in his baricenter $B$. For the ski, we use the
model considered by Aldo Bressan (1991), with a special approximation of the
skier. Our main concern is the existence and the structure of time-optimal
controls. As first step, we introduce an auxiliary control system with control
entering linearly and show the relationships between this and the original
system. We then perform a detailed study of the auxiliary system, using
geometric techniques. In turn this provides accurate information on the
time-optimal controls for the original system.
