Gaining a mathematical edge in the Tour de France
Ian Griffiths, Royal Society University Research Fellow at the Mathematical Institute
We are currently in the middle of one of the most intense competitions of physical endurance: the Tour de France. This year’s route covers a total of 3351km over 23 days with only two of these rest days. Those who have been following the Tour so far will be familiar with the often predictable format, with the majority of riders cycling together as a main group, or peloton. The reason for this behaviour is that cycling in a group reduces the air resistance that is experienced by the cyclist. With energy savings of up to around a third in the peloton compared with riding solo, it is highly energetically favourable to stay with the main field.
However, if a cyclist wishes to win a Tour stage then they must at some point make a break from the peloton. A sprinter, such as Mark Cavendish or the current leader, Julien Bernard, will remain in the peloton until just a few hundred metres from the finish line, knowing that they have the power to beat the majority of riders in a one-on-one sprint finish. For the remainder of the group though, this leaves a dilemma. If they are to win a stage then they must break away far enough from the finish line to ensure the sprinters do not triumph. But in doing they must leave the peloton and the advantage this offers against the wind resistance.
It is this requirement that leads to the common occurrence of a rider (or sometimes a small group of riders) making a break from the peloton relatively far from the finish line. We watch in admiration as this lone rider puts everything they have into maintaining a separation from the main field, battling against the added wind resistance. As the finish approaches, the peloton will organize itself into a streamlined unity to reel the rider back in, often occurring despairingly close to the finish line, with all the efforts of the lone rider coming to nothing.
How can we avoid these moments so that a lone cyclist ultimately triumphs? Mathematics can offer one possible strategy by framing the scenario as an optimization question. A lone cyclist will not be able to sustain the extra force required to cycle outside of the peloton indefinitely, with fatigue effects coming into play. So, if the cyclist breaks away too soon then they risk fatigue effects striking before the finish line and being caught by the peloton. On the other hand, if the cyclist breaks too late then they reduce their chance of a large winning margin. Posing this generally, we ask the question: ‘for a given course profile and rider statistics, what is the optimum time to make a breakaway that maximizes the finish time ahead of the peloton?’.
To answer this question, a mathematical model is derived for the cycling dynamics, appealing to Newton’s Second Law, which captures the advantage of riding in the peloton to reduce aerodynamic drag and the physical limitations (due to fatigue) on the force that can be provided by the leg muscles. The effect of concentration of potassium ions in the muscle cells is also a strong factor in the fatigue of the muscles: this is responsible for the pain experienced in your legs after a period of exertion, and is what sets a rider’s baseline level of exertion. The model derived captures the evolution of force output over time due to all of these effects and is applied to a breakaway situation to understand how the muscles respond after a rider exerts a force above their sustainable level.
Asymptotic techniques are then used, which exploit the fact that the course may be divided into sections within which variations from a mean course gradient are typically small. This leads to analytical solutions that bypass the need for performing complex numerical parameter sweeps that might otherwise make the analysis prohibitively expensive. The asymptotic solutions crucially provide a method to draw direct relationships between the values of physical parameters and the time taken to cover a set distance.
The model serves to frame intuitive results in a quantitative way. For instance, it is expected that a breakaway is more likely to succeed on a climb stage, as speeds are lower and so the energy penalty from wind resistance when cycling alone is reduced. The theory confirms this observation while also providing a measure of precisely how much more advantageous a breakaway on a hill climb would be. For multiple stage races the theory can even identify which stages are best to make a breakaway and when it is better to stay in the peloton for the entire stage to conserve energy. For multiple-rider breakaways the riders can switch turns on the front so that they are still able to shield themselves from the wind and conserve energy for periods of time. As a result, multi-rider breakaways have a higher chance of success. The theory can also be adapted to address such scenarios, providing guidance on how many riders would be needed to form a breakaway group that would sustain a lead over the peloton.
The resulting theory could in principle allow a cycle team to identify the strategy and exact breakaway position during each stage in advance of a major race, with very little effort. Last year’s Tour de France was won by a margin of less than one minute for a total race time of more than 86 hours. As a result, the prior information offered by these mathematical models could provide the necessary edge to secure the required marginal gains.
It is clear that winning a Tour de France stage involves a great deal of preparation, physical fitness and, ultimately, luck on the day. However, coupled with sports scientists, engineers, and dieticians, mathematics can provide a fundamental underpinning for the race dynamics that can guide strategies to increase the chance of such wins.
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