Calculus and Elite Running Pace Strategy
Running a world-record track and field race depends on a lot of factors: psychological, physiological, biological, and physical. Perhaps not surprisingly, underlying much of it, is the mathematical.
The mathematical analysis of running dates back to Joseph Keller's "A theroy of competitive running". In Keller's classic paper, he models the human body's physiological capacity for running with set of simple differential equations and arrives at some very interesting results.
If you've ever been in a race longer than 200m, you've probably given at least some thought as to how hard you should run throughout the race. Should you sprint all out in the beginning or save your energy for the finish? Keller's mathematical model actually provides some insight into this question. His model predicts that for all races under 291 meters, the optimum strategy is to sprint at 100% acceleration for the entirety of the race. Races longer than 291 meters require a more complicated strategy to optimize performance.
For example, long distance runners usually try to maintain moderate constant speeds throughout most of the race and finish with a "kick". However, elite mid-distance runners—running the 400m and 800m races—have long known that running the first half the race faster than the second half actually leads to the faster possible overall times. Keller's model can't account for these strategies, but there do exist compelling models for both of these cases (for example, see Whitt [2009] for long distace race strategy; see Reardon [2012] for mid-distance strategy)
Of course, the most important factors in elite running are likely non-scientific and rely more on human experience and mental fortitude. However, as we've seen in many other sports thought to be immune to the analysis of lofty academics (e.g., baseball, soccer, etc.), running is—slowly but surely—also being transformed by the careful application of mathematical and scientific insights.
