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Brachistochrone on a Velodrome

The Brachistochrone problem, which describes the curve that carries a particle under gravity in a vertical plane from one height to another in the fastest time, is one of the most famous studies in classical physics. There is a similar problem in track cycling, where a cyclist aims to find the trajectory on the curved sloping surface of a velodrome that results in the minimum lap time. In this paper we extend the classical Brachistochrone problem to find the optimum cycling trajectory in a velodrome, treating the cyclist as an active particle. Starting with two canonical cases of cycling on a sloping plane and a cone, where analytical solutions are found, we then solve the problem numerically on the reconstructed surface of Velodrome de Montigny le Bretonneux in France.

The Brachistochrone problem, which describes the curve that carries a particle under gravity in a vertical plane from one height to another in the fastest time, is one of the most famous studies in classical physics. There is a similar problem in track cycling, where a cyclist aims to find the trajectory on the curved sloping surface of a velodrome that results in the minimum lap time. In this paper we extend the classical Brachistochrone problem to find the optimum cycling trajectory in a velodrome, treating the cyclist as an active particle. Starting with two canonical cases of cycling on a sloping plane and a cone, where analytical solutions are found, we then solve the problem numerically on the reconstructed surface of Vélodrome de Montigny le Bretonneux in France. Introduction.−In 1696 Johann Bernoulli posed a problem to the scientific community which, after a year and half, had only been solved by a handful of individuals, including Newton and Leibniz [2,3]. Along with Newton's minimal resistance problem, it was one of the first mathematical studies that pioneered the field of the variational calculus, and so had an immense influence thereafter [4]. The Brachistochrone problem, whose etymology comes * graham.benham@ladhyx.polytechnique.fr from the ancient Greek for shortest time [2], describes a curve that carries a particle under gravity in a vertical plane from one height to another in minimal time. The solution is equivalent to the path traced out by a rolling circle, also known as a cycloid. Other variations of this problem have included the effects of friction [5][6][7], the motion of a disc on a hemisphere [8], and even the quantum Brachistochrone problem [9].
A natural extension of the Brachistochrone problem is the motion of a cyclist in a velodrome. In track cycling, athletes compete to move around the sloped velodrome surface as fast as possible, much like a higher dimensional Brachistochrone. Whilst there are many different types of velodrome races, the type of race that lends itself most obviously is the qualification time trial [10]. In this case cyclists complete three and a half laps of the velodrome, where only the time for the final 200 m of the last lap is measured. The cyclists build up speed on a single high gear over the first two and half laps [11], staying as high up on the velodrome slope as possible to maintain large potential energy. Then, on the final lap they descend the slope and sprint around the track as fast as they possibly can. Similarly to the Brachistochrone, the choice of the descent trajectory is critical. Time is lost if the descent trajectory is too steep or too shallow. The optimal trajectory must balance the exchange between potential and kinetic energy perfectly.
In Fig. 1(a-d) we show four early time snapshots taken from a velodrome qualification race at Vélodrome de Montigny le Bretonneux, France. As the cyclist enters the third and final lap (at an initial speed of around 57 km/hr), they descend into the sharp corner of the velodrome over a period of around 1 s. After the descent, cyclists typically remain near the bottom of the slope, within around 1 m of the blue côte d'azur lane (see Fig.  1(d)), for the duration of the final lap. Hence, the initial descent is of critical importance because it is the only period of the race where variation is observed between different cyclists. In Fig. 1(e) we also display the final times of the top five cyclists from the Rio 2016 Olympic games. The difference in time between cyclists is usually around the order of one tenth, and sometimes one hundredth of a second (corresponding to 0.1 − 1% of the total time). Hence, if the track time can be reduced even slightly by choosing a better descent trajectory, this can have a significant impact on the final ordering of the athletes.
In this study we show how to find the optimum trajectory of a cyclist in a velodrome by modifying the classical formulation of the Brachistochrone problem, treating the cyclist as an active particle on a surface. Since the velodrome track is naturally decomposed into straight and curving sections, we start by studying the two canonical cases of motion of a cyclist on a plane and a cone, for which analytical solutions can be found using the Euler-Lagrange equations. Then, using geometrical data taken from the Vélodrome de Montigny le Bretonneux track, we reconstruct the velodrome surface and solve the corresponding optimisation problem using a numerical method that extends from the previous examples.
Brachistochrone on a plane.−Before discussing motion on two-dimensional surfaces, let us first summarise the classic Brachistochrone problem, formulated in the Euler-Lagrange setting [2,12].
Consider a particle of mass m that moves in the vertical plane (x, z) under gravity g. This is equivalent to motion on a two-dimensional plane in the case where the plane makes an angle α = π/2 with the horizontal, where α is illustrated in Fig. 2(a). We seek to minimise the total time for the particle to move along a trajectory from position (0, 0) to (L, −H), which is given by where s is the arclength of the trajectory, varying from 0 to s 0 , and v = √ẋ 2 +ż 2 is the speed of the particle. Neglecting friction, the total energy of the particle is conserved, such that where we have assumed that the particle is initially at rest. Hence, by using (2) and by rewriting (1) in terms of x and z, where ds = dx 1 + (dz/dx) 2 , the total time is This quantity (3) can be minimised by solving the Euler-Lagrange equation for the function z(x), which is 2z together with the boundary conditions In the case where the final height −H is included as an optimisation variable, the boundary condition (6) is replaced by It is well-known that the solution is a cycloid which, in the case of the latter boundary condition, is given parametrically by To extend the above formulation to motion on a sloping plane is relatively straightforward. Let us now consider a coordinate system (x, y, z), in which the particle moves on a plane z = y tan α that makes a constant angle α with the horizontal. By considering the rotated coordinate Y = y/ cos α that lies in the plane, it follows that the total energy is given by Hence, the total time for the descent is which is equivalent to the classic Brachistochrone but with a modified gravity g = g sin α, and a trajectory Y (x) that lies in the plane. Hence, the parametric solution is the same as before (8), except with z replaced by Y , and a total descent time multiplied by a factor of 1/ √ sin α. Brachisto-'cone'.−The next most canonical case of a two-dimensional surface is a cone. This is of particular interest to track cycling because the sharp corner at each end of the velodrome is approximately conical, as we discuss later. By converting the above formulation to cylindrical polar coordinates, it is possible to write down the energy equation for motion on the surface of a cone, which is where R = r/ cos α is the rotated radial coordinate, and R 0 is the initial position of the cyclist. Unlike the planar case, where the initial position is irrelevant, in the conical case R 0 is a necessary parameter, and is related to the curvature at the initial position. The resulting timeminimisation problem is written in terms of the integral where θ 1 − θ 0 is the angle traced out by the trajectory. The resulting Euler-Lagrange equation for the trajectory R(θ) is with boundary conditions The Brachisto-'cone' problem (13) is different from the Brachistochrone problem (4) since it also takes into account the effects of rotation, such as the centrifugal force. Such forces play an important role in track cycling, because they allow the cyclist to perform tight corners at high velocity by tilting their bike dramatically in the direction of the bend.
Owing to the more complicated form of (13) neither an explicit nor a parameterised closed form solution is available (though one constant of integration can be found by considering the Beltrami identity). However, we can solve the boundary value problem (13)-(15) numerically. In the subsequent sections, we refer to the solutions in the two cases of the planar Brachistochrone and the Brachisto-'cone' as analytical, but indeed only the planar case is in closed form.
Cycling Brachistochrone.−By replacing the particle in the above examples by a point cyclist (or an active particle), we see that the optimal curves are exactly equivalent to the optimal cyclist trajectories in the case where the cyclist only uses a leaning force to manoeuvre instead of pedalling (i.e. conservative motion). If we consider that the cyclist may also apply a pedal thrust in the direction of motion and experiences wind drag, then the total energy is no longer constant. In this case, the dynamics of the cyclist, which we derive in the Supplemental Material [13], are given by where F ⊥ is the leaning force perpendicular to the direction of motion, and F is the force parallel to the direction of motion, which is divided into a pedal thrust and a drag force F = F p − F d . We model the drag using the parameterisation F d = 1/2ρC d Av 2 , where C d is the drag coefficient, A is the combined surface area of the cyclist and the bicycle, and ρ is the density of air [14][15][16][17][18]. Note, the only contribution from the normal force is the gravity term in (17). The total energy of the system satisfies d dt Clearly, if we set F = 0 then (18) leads to the former Euler-Lagrange formulation. If F = 0 then we cannot solve the problem analytically, but a numerical solution can be found which we discuss shortly. For the case of the Brachisto-'cone' with forcing, the energy equation is identical to (18), except with Y replaced by R.
Physiology.−To model the pedalling force F p (t), there are certain mechanical and physical considerations that must be taken into account. In particular, since track cyclists must choose a fixed gear ratio for the duration of the race, the pedalling force depends strongly on the instantaneous pedalling rate, and this relationship depends on the physiology of the individual cyclist.
As shown by Dorel et al. [10], the pedal torque that a cyclist applies in a sprint is a linearly decreasing function of the pedalling frequency. This linear relationship is characterised by two coefficients T max and ω max , which correspond to the maximum possible torque (occurring at zero pedalling frequency) and the maximum possible pedalling frequency (occurring at zero torque). Each cyclist has a sprint performance characterised by these two parameters, and these are easily measured with a pedalling experiment.
The pedalling torque and frequency are related to the pedalling force F p and speed v via the development D, which is the distance travelled by one single rotation of the pedals (analogous to the gear). Hence, the pedalling force is given by the linear relationship Note that (19) is similar to the force-velocity relationship related to muscle physiology, sometimes called the Hill equation [19]. However, in the case of the Hill equation there is an additional denominator of the form (1 + v/Dω max ).
In Fig. 2 we plot solutions to both the planar Brachistochrone and the Brachisto-'cone' problem. Black curves correspond to the analytical solution in the case of zero pedalling and drag force F = 0. Light blue dashed curves correspond to the numerical solution to the equivalent optimal control problem, which is achieved by formulating an interior point constrained optimisation [20,21] using the dynamics (16)- (17) to govern the variables x(t), y(t), and using the forcing F ⊥ (t) as a control function (see the Supplemental Material [13]). The analytical solution is useful for validating the numerical approach, giving us confidence when applying it to the case of non-zero pedalling and drag force, for which an analytical solution is not available. Such solutions (for F = 0) are shown on the same plot with dotted blue curves.
To calculate these trajectories we choose values for the model parameters that correspond to realistic cycling scenarios. We take the combined mass of the cyclist and the bike as m = 86 kg, and the Dorel curve characteristics ω max = 250 rpm, T max = 230 Nm, correspond to data taken from an elite athlete [10]. The product of the drag coefficient and the surface area is C d A = 0.22 m 2 , which is equivalent to a streamlined cycling position and modern equipment [15][16][17][18][22][23][24]. Finally, we choose a development of D = 8.5 m, which is typical for velodrome track racing.
From Brachistochrone to velodrome.−The next step is to apply the above method to find the optimum trajectory on a real velodrome track. As a case study, we choose the Vélodrome de Montigny le Bretonneux in France. For this velodrome, the inside lane is composed of two straight lines of length L = 38 m, connected by two half-ellipses of semi-major and semi-minor axes a = 29.8 m and b = 24.2 m (see Fig. 3(c)). Since the velodrome slopes in the normal direction with an angle that varies as one moves around the track, it is convenient to make use of the tangent and normal coordinates measured on the inside lane (s, n) (see the schematic diagram in Fig. 3(a)). In terms of these coordinates, the velodrome surface is written simply as z = n tan α(s), where the width of the surface is a constant W = 7.9 m. Therefore, unlike the previous examples, motion on the velodrome is bounded, such that 0 ≤ n ≤ W cos α. In Fig. 3(b) we plot the angle α(s), which was measured at the race track using an angle-metre. The slope angle is approximately sinusoidal, varying between 14 • in the middle of the straight sections to 45 • in the middle of the curved sections. The reconstructed velodrome surface is shown in c).
Based on the previous canonical examples, we formulate and solve a numerical optimisation problem for the trajectory, where the dynamics in the straight regions are similar to (16)- (17), and for the elliptical sections of the velodrome we consider a small perturbation from the cone example, where the ellipticity = a/b − 1 = 0.23 is treated as a small parameter in an an asymptotic expansion (see the Supplemental Material [13]).
In Fig. 4 we display the optimal trajectory, as well as plots of velocity v and power P = vF p as functions of time  Fig. 1(a-d). We also indicate the different curved and straight regions of the velodrome, as well as the start and finish lines for the final 200 m of the track. (b) Corresponding speed v and power P profiles for the Brachistochrone, compared with measured data from an elite athlete. Straight regions of the track are illustrated with shading. [25]. For this example, we use the same parameter values as before, except with an initial velocity of 57 km/hr purely in the x-direction, which corresponds to the typical speed before descent for Olympic athletes. The optimal trajectory is very similar to the strategy employed by athletes, descending before the first corner and then hugging the inside lane thereafter. The descent follows closely to the position of the cyclist in the snapshots in Fig. 1(a-d), which we indicate approximately with white stars. The velocity and power in Fig. 4(b) are compared to data taken from an elite cyclist in the Vélodrome de Montigny le Bretonneux, showing close agreement. This indicates that the cyclist's trajectory is already nearly optimal, and illustrates the validity of our model. The time to complete the final 200 m of the lap is around 10 s, which is close to high-ranking Olympic performance, as seen in the table in Fig. 1(e).
Concluding remarks.−Whilst the classic Brachistochrone problem is limited to one dimension, we have shown how a similar formulation can extend the problem to motion on a plane or a cone, for which the Euler-Lagrange equations yield analytic solutions.
By extending the classic Brachistochrone using these two canonical cases, as well as including the effects of cyclist physiology, we have shown how to optimise the trajectory of a cyclist in the real example of the Vélodrome de Montigny le Bretonneux, finding very close agreement with measured athlete data. This work not only paves the way for future studies of cycling, but also has implications for research across the field of physics on the topic of active particles on surfaces .
We thank Tullio Traverso and Benjamin Lallemand for useful discussions. We gratefully acknowledge the support from Ecole Polytechnique for the research program Sciences 2024.