Using calculus of variations to optimize paths of descent through ski race courses

Abstract

The goal of ski racing is to pass through a series of gates as quickly as possible. There are many paths from gate to gate, but there is only one path that is fastest. By knowing what the fastest path is, a racer could shave tenths of seconds off his or her time. That is a tremendous amount of time considering that races are won by hundredths of a second. This thesis attempts to calculate the fastest path through a ski race course using several simplifications such as neglecting friction. The method of attacking this problem is to modify the Brachistochrone problem. It is found that it is best if the skier places the apex of the turn at the gate, and that turning more after the gate is better than turning more above the gate. In the case of a rhythmical course, it is found that turning more below the gate is still true, but not as evident. Instead the optimal path appears more symmetric about the gate.