## Mathematical constraints of Backus averaging

## Mathematical constraints of Backus averaging

## Mathematical constraints of Backus averaging

## Mathematical constraints of Backus averaging

## Mathematical constraints of Backus averaging

## VAM

To maximize the mean ascent speed, one needs to choose an appropriate hill.

With our mathematical models we show that, for a constant power, the maximum is achieved by riding along as steep of a hill as possible.

Using the calculus of variations, we also show that the inclination should be as steady as possible.

To optimize VAM, we examine it as a function of the cadence and gear ratio.

Recently established records provide an empirical support for these analytical results.

These results constitute a mathematical-physics background upon which various strategies for the VAM maximization can be examined. In particular, in the context of physiological considerations, one could consider the maximum sustainable power as a function of both the force applied to the pedals and their circumferential speed. For instance, one might choose a less steep slope to allow a higher cadence x gear-ratio product if it allows to generate and sustain a higher power.

For everesting, one might consider a very short and exceptionally steep slope, while keeping in mind the issue of measurement errors due to the shortness of the segment itself. In all cases, however, the slope needs to be constant.

Illustration of the concept of maximum steepness and steady incline, by Roberto Lauciello (2020)

## REFERENCE PUBLICATIONS

Bos, L., Slawinski, M.A., Stanoev, T. (2021) On maximizing VAM for a given power. Slope, cadence, force and gear-ratio considerations, 2006.15816v5 [physics.pop-ph]

Bos, L., Slawinski, M.A., Stanoev, T. (2020) On maximizing VAM for a given power: Slope, cadence, force and gear-ratio considerations, 2006.15816 [physics.pop-ph]