Sure, we can become better runners by hydrating well, eating right, cross training, and practice. But getting to an optimal running strategy with equations? That's exactly what a pair of mathematicians from France propose in a paper published this month in the SIAM Journal on Applied Mathematics .
The model uses a system of ordinary differential equations. Aftalion explains: "Our model relies on two basic principles: energy is preserved, and acceleration (or variations of velocity) is equal to the sum of all forces. This leads to a system of differential equations coupling the unknown variables of the runner (velocity, propulsive force and anaerobic energy), and dependent on physiological parameters such as maximal oxygen uptake and total available anaerobic energy."
"The difficulty in numerical simulation is that the equations are coupled, hence, none of the variables can be solved independently of the others. Using an optimal control solver developed by Inria (The French Institute for Research in Computer Science and Automation), we are able to get a full numerical solution, something that hasn't been done previously," says Aftalion.
Performance data from races helps in assessing this new model. "We are now able to identify the physiological parameters of a person through data of a good race on a given distance using several time measurements at regular distances in the race," says Aftalion. "From this we can predict how to run an ideal race, both for a champion, helping him improve his performance and win a medal, as well as for a regular runner who lacks professional coaching and seeks help. Our predictions corroborate actual strategies used by professional athletes."
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Source Article
Optimization of running strategies based on anaerobic energy and variations of velocity SIAM Journal on Applied Mathematics , 74(5), 1615–1636 (Online publish date: October 9, 2014). The paper is available for free download at the link above until January 27, 2015.
About the Authors
Amandine Aftalion is Research Director at Centre national de la recherche scientifique (CNRS) in Paris, France, and Frederic Bonnans is a research team leader at Ecole Polytechnique in Palaiseau, France.
If you would like to schedule a direct interview with the paper's authors, please email pao@siam.org .
SIAM Journal on Applied Mathematics