On the brachistochrone problem for cycling ascents

TL;DR

The paper demonstrates that for cycling ascents with fixed start and end points and power constraints, the optimal trajectory minimizing ascent time is a straight line at constant speed, contrasting classical gravity brachistochrone solutions.

Contribution

It establishes that the minimum-time climbing path under power constraints is a straight line with constant speed, extending classical brachistochrone results to cycling scenarios.

Findings

Optimal ascent path is a straight line at constant speed.

Constant power along the path minimizes ascent time.

Contrasts with classical gravity brachistochrone solutions.

Abstract

VAM ({\it velocit\`a ascensionale media}) is a measurement that quantifies a cyclist's climbing ability. We show that to minimize the time to attain a given height gain\, -- \,which is tantamount to maximizing VAM\, -- \,a cyclist should climb as steep a constant-grade hill as possible. Apart from the power-to-weight ratio, the limit of steepness is imposed by such factors as the efficiency of pedalling, which is related to feasible cadence, maintaining balance, preventing lifting of the front, and skidding of the rear, wheel. In an appendix, we discuss steepness constraints due to pedalling efficiency. The article itself is focused on consequences of the power available to the cyclist, which can be viewed as a necessary condition to examine other aspects of climbing strategy. We show that\, -- \,for given start and end points, and for any fixed average-power constraint\, -- \,the brachistochrone, which is the trajectory of minimum ascent time, is the straight line connecting these points, covered with a constant speed, which along such a line is equivalent to a constant power. This is in contrast to the classical solution of a descent brachistochrone under gravity, which is a cycloid along which the speed is not constant.