Using Mathematics To Improve Running

Researchers with the French National Center for Scientific Research, or CNRS (Centre national de la recherche scientifique), have created a mathematical model that seeks to improve a runner’s performance, weight and fitness. Their findings, which could conceivably result in personalized e-coaching applications leveraging an individual’s physiological state, conclude that runners who vary their speed are better able to conserve energy and therefore run longer.

The mathematical model uses differential equations to combine speed, acceleration, propulsion forces and friction, maximum oxygen uptake (VO2 max) and anaerobic energy coupled with initial conditions (zero speed and fixed energy) and constraints (positive energy and propulsion force) to predict an optimal running strategy for a race. In addition to determining that by varying speed runners conserve energy and run longer, the model also allows researchers to identify which physiological factors a runner should modify in order to improve performance.

The researchers are hoping to develop software can could serve as a personal e-coach and are at the same time looking to incorporate additional parameters such as altitude and wind speed/direction into the model so that it can be applied to other sports beyond running like cycling or swimming.

I would love to see an iPhone app for this, or even better incorporation of the model into an existing app like RunKeeper. Apart from helping with training it would be great during an actual race, for example outlining the best plan to complete the second half of a marathon in which I’ve expended too much energy during the first half.

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