On control problems of minimum time for Lagrangian systems similar to a swing. I. Convexity criteria for sets

The authors consider holonomic systems with two degrees of freedom, one of the variables being a control variable, and develop a general theory. They study problems of a swing of variable length and the motion of a pair of skis when friction and air resistance are neglected, and construct an appropriate variational principle for the time-optimal control. The existence of a solution is shown to be associated with the convexity of a set involving the configuration coordinate and the control variable. Suggestions are made as to how a skier should behave under various circumstances.