Long-distance cycle races, such as a Tour de France stage, typically follow a prescribed pattern: riders cycle together as a main group, or peloton, for the majority of the race before a solo rider, or small group of riders, makes a break from the peloton, usually relatively close to the finish line. The main reason for this behaviour is that cycling in a group reduces the air resistance that is experienced by a cyclist. With energy savings of up to around a third when cycling in the peloton compared with riding solo, it is energetically favourable to stay with the main field for the majority of the race. However, if a cyclist wishes to win a race or a Tour stage then they must decide on when to make a break. In doing so, the rider must provide an additional pedal force to offset the effects of air resistance that would otherwise be mitigated by riding in the peloton. However, the cyclist will not be able to sustain this extra force indefinitely, with fatigue effects coming into play. As a result, a conflict emerges: if the cyclist breaks away too soon then they risk fatigue effects kicking in 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.

So Oxford Mathematicians Ian Griffiths and Lewis Gaul and Stuart Thomson from MIT asked 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 the 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 you experience 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 used that 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. The asymptotic solutions 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. The resulting theory could 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. Such prior information could provide the necessary edge required to secure the marginal gains required to win a race.

While it is clear that winning a Tour de France stage involves a great deal of preparation, physical fitness and, ultimately, luck on the day, mathematics can provide a fundamental underpinning for the race dynamics that can guide strategies to increase the chance of such wins.

*Story courtesy of the Mathematical Institute*