## 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

## POWER AND RESISTANCE

Isaac Newton subject to the effects of gravity, by Roberto Lauciello (2020)

Power-meter measurements together with GPS data are now a common way to estimate the performance of a cyclist.

They can also be used during a ride to estimate the coefficients of the air, rolling and drivetrain resistance without the need for a wind tunnel or a laboratory.

In this section of our work, we relate power-meter measurements to a mathematical model to examine the conversion of power generated by a cyclist into a motion of a bicycle. In particular, we infer the values of the air, rolling and drivetrain resistance by seeking an acceptable agreement between measurements and model.

To do so, we combine classic formulations of fluid mechanics with innovative optimization methods. We also invoke aspects of approximation theory to comment on the empirical adequacy of estimated values.

## RELATIONS AMONG MEASUREMENTS

We examine the effects of various quantities on performance and determine their relative importance; for instance, what is the importance of weight versus air resistance, and how it changes depending on speed and inclination.

To address such questions, we invoke the implicit function theorem.

## TIME-TRIAL STRATEGIES

How to optimize the effort of a time trial with segments with and against the wind? For instance, should the power or the speed be kept constant?

To address such questions, we invoke the Lagrange multipliers.

Joseph-Louis Lagrange with and against the wind, by Roberto Lauciello (2020)

## REFERENCE PUBLICATIONS

Danek, T., Slawinski, M.A., Stanoev, T. (2020) On modelling bicycle power-meter measurements: Part I. Estimating effects of air, rolling and drivetrain resistance. 2005.04229 [physics.pop-ph]

Danek, T., Slawinski, M.A., Stanoev, T. (2020) On modelling bicycle power-meter measurements: Part II. Relations between rates of change of model quantities, 2005.04480 [physics.pop-ph]