Brachistochrone on a Velodrome

C Clanet; C Cohen; E Brunet; GP Benham

REVIEW 4 major objections 4 minor 29 references

Brachistochrone on a Velodrome

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Extending the brachistochrone to an active cyclist on a banked track yields a near-optimal elite descent and a final 200 m time near 9.7 s.

desk verdict A genuinely new application of brachistochrone methods to track cycling, with a plausible model and real athlete data, but the near-optimality claim outruns what is actually checked. read the letter →

arxiv 1908.02224 v2 pith:LM4TGFJN submitted 2019-08-06 physics.flu-dyn

GP Benham , C Cohen , E Brunet , C Clanet This is my paper · ORCID

classification physics.flu-dyn

keywords brachistochronetrackcyclingvelodromeactiveparticlesoptimalcontrolaerodynamicdragpedallingtorquefatigue

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends the classical brachistochrone problem to track cycling by treating the cyclist as an active point particle that pedals against aerodynamic drag on a banked surface. It first solves the minimum-time descent analytically for a sloping plane and a cone, then reconstructs a real velodrome's surface and finds the fastest trajectory numerically with the sideways force as the control. The computed speed and power curves closely track data from an elite rider, which the paper reads as evidence that the observed descent is already nearly optimal. The model also predicts a final 200 m time of about 9.7 s, in line with top Olympic qualifying performances, and explains the riders' choice of descent start via a fatigue timescale.

What carries the argument

The central object is the active-particle brachistochrone: a point cyclist on a surface $z=f(x,y)$ with a pedalling force $F_\parallel$ and a perpendicular steering force $F_\perp$, so the energy changes according to $dE/dt = vF_\parallel$. The pedalling force comes from a linear torque--cadence relation $F_p = (2\pi T_{\max}/D)(1-2\pi v/(D\omega_{\max}))$, drag is quadratic $F_d=\tfrac{1}{2}\rho C_d A v^2$, and the track is encoded as $z=n\tan\alpha(s)$ with constant width $W=7.9$ m, the curved ends treated to first order in the ellipticity. The optimization discretizes the state variables in time, keeps the rider inside the track with barrier constraints, and solves the minimum-time problem with the perpendicular force as control, using a penalty and interior-point scheme; the planar and conical analytical solutions validate the numerics.

What would settle it

Recompute the same optimum on a velodrome represented with exact elliptical ends rather than the first-order expansion, and add a constraint that the normal reaction at the surface stays positive, a grip limit; if the fastest line or the final 200 m time changes by more than a few hundredths of a second, the close agreement with athlete data would be shown to be non-decisive. Alternatively, timed trials in which a rider deliberately holds a measurably different lateral line and still matches the predicted 9.7 s would indicate the model is missing a relevant constraint.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the classical brachistochrone, generalized to an active particle on the measured surface of a velodrome, reproduces the elite descent closely enough to conclude that the athlete's trajectory is already nearly time-optimal. The surface is written as $z=n\tan\alpha(s)$ in tangent and normal coordinates on the track's inside lane, with the measured slope angle $\alpha(s)$ varying from about $14^\circ$ on the straights to about $45^\circ$ in the banks. Solving the time-minimization problem with pedalling torque that decreases linearly with cadence and quadratic aerodynamic drag yields an optimal line that descends before the first corner and then hugs the inside lane, with a final 200 m time near 9.7 s and velocity and power profiles that match measurements from an elite rider.

Load-bearing premise

The result depends on the measured slope, initial speed, and surface geometry being exact, on the curved ends being adequately captured by a first-order expansion in the 23% ellipticity, and on the rider being able to apply any sideways force without grip or balance limits; if any of these fails, the calculated fastest line may not be a line a real rider can hold.

Editorial extensions

If this is right

The descent phase is the only part of a qualifying lap where rider-to-rider variation is observed, so the fractions of a second that separate medal positions come from the line chosen there; the model shows where those fractions come from.
For a given track geometry and rider physiology, the optimal line can be computed in about 10 seconds on a laptop, making on-site tactical optimization feasible.
The optimal start of the descent is set by the fatigue balance: peak speed should occur at the finish line, and observed starting positions from 3 m to -11 m correspond to sprint times of 13-15 s and peak speeds around 74-75 km/h.
Riders should start from the top of the banking, because the optimal initial normal position is always $n(0)=W\cos\alpha(s(0))$, maximizing potential energy before the descent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One testable extension is to vary rider mass, gear development, or drag coefficient in the same solver; the model should predict how the optimal descent line and final 200 m time shift, which could be checked against athletes with different body types.
The claim of near-optimality rests on a single rider's data on one velodrome; a stronger test would record several riders' trajectories on the same track and see whether their spread brackets the computed optimum.
Because the curved ends are modelled only to first order in ellipticity and the steering force is unbounded, recomputing with an exact ellipse and a grip-limited normal reaction would show whether these simplifications matter at the hundredths-of-a-second level that decides races.
If the same framework were extended to team pursuit, where slipstreaming changes the drag force, the optimal individual line might differ from the time-trial line, giving teams a quantitative tool for tactics.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth a read if you care about sports physics or active-particle optimal control. The new thing here is taking the classic brachistochrone problem, adding a pedalling force and drag, and solving the minimum-time line on a real velodrome from measured surface geometry. That is a real step beyond the toy problems. The plane case reduces to a cycloid with rescaled gravity, which is fine. The cone case is not closed-form despite the paper calling both 'analytical'—the text is honest about that, so it's a labelling wart, not a deception.

The validation is the strongest part. The model parameters come from independent physiology and wind-tunnel sources, not fitted to the athlete traces. The computed trajectory reproduces the qualitative descent strategy and the speed and power curves match three elite attempts quite well. Given how small the time margins are in sprint cycling, a tool that identifies the descent line is practically useful.

The soft spots: First, Eq. (3.2) as printed has R0−R in the denominator with no square root, which is dimensionally wrong. The derived Euler-Lagrange equation (3.3) corresponds to the sqrt version, so this looks like a typo, but it is exactly the kind of typo that confuses readers. Second, the optimisation treats the lateral force F_perp as unbounded. There is no grip or normal-reaction constraint. That matters because the paper's headline claim—that the athlete's trajectory is already nearly optimal—requires the computed optimum to be feasible. If the unconstrained line needs lateral accelerations no rider can produce, the computed 'optimum' is just a lower bound, and the agreement with data doesn't by itself certify near-optimality. The authors don't report F_perp, so you can't check. Third, the elliptical curved sections are treated to first order in epsilon=0.23, which is not tiny; the surface may be off by more than the claimed optimality margins. Finally, no data or code is included, so the comparison with athlete data can't be independently audited.

None of these sink the paper. The model is transparent, the biology and aerodynamics are standard, and the application is genuinely new. But the near-optimality conclusion is stated more strongly than the constraints justify. If the authors report the required F_perp and verify it is within a rider's capability, or add a constraint and show the optimum barely changes, the central claim would be solid.

Who is it for: sports physicists, control people, and cycling biomechanists. I'd bring it to a reading group and I'd probably cite it. It deserves proper peer review, with the lateral-force question as the main referee ask.

Referee Report

4 major / 4 minor

Summary. The paper extends the classical brachistochrone problem to track cycling by treating a cyclist as an active point particle on a sloping surface. It first gives analytical Euler-Lagrange solutions for motion on a plane and on a cone, then solves a numerical optimal-control problem on a reconstructed model of the Montigny-le-Bretonneux velodrome. The authors compare the predicted speed and power profiles with elite-athlete measurements and conclude that the athlete's trajectory is already nearly optimal, with a final-200 m time of about 9.7 s. A separate discussion addresses the choice of initial conditions and the role of fatigue.

Significance. If the central claims hold, this is a useful and interesting contribution: it generalizes a classical variational problem to active particles on curved surfaces and provides a practical tool for optimizing track-cycling lines. The paper has clear strengths: the physiological and aerodynamic parameters are taken from independent published sources rather than fitted to the validation data; the canonical planar and conical cases provide testable analytical benchmarks; the numerical implementation is modern and reproducible in principle; and the fatigue scaling leading to Eq. (6.2) gives a simple falsifiable prediction. However, the validation claim that the athlete's trajectory is 'already nearly optimal' rests on several modeling choices that are not quantitatively checked, in particular the absence of any bound on the lateral control and the first-order treatment of the elliptical track geometry. These issues do not invalidate the overall approach, but they must be addressed before the main claim is fully supported.

major comments (4)

[Section 3] The printed Eq. (3.2) has an integrand with denominator R0-R, which is dimensionally inconsistent with the energy equation (3.1). From (3.1), the speed satisfies v^2 = 2g sinα (R0-R), so the time functional should contain the square root of R0-R in the denominator. Since the reported Euler-Lagrange equation (3.3) appears to correspond to the corrected functional, Eq. (3.2) should be corrected, and the derivation should be checked for consistency.
[Appendix B, Section 6a] The numerical optimization in Appendix B treats the lateral force F_perp as an unbounded control, subject only to the track-width inequalities in (A4). No bound on |F_perp|, no normal-reaction constraint, and no tyre-friction limit is imposed. Because F_perp does no work, the optimizer can in principle generate arbitrarily tight corners without affecting the longitudinal speed profile, so the computed line may be infeasible for a real cyclist. The paper should report the optimal F_perp profile and verify that the required lateral acceleration is within the grip and banking limits of the track. Without this check, the agreement in Fig. 5 does not establish that the athlete's trajectory is near-optimal.
[Section 6a, Appendix A(c)] The reconstructed velodrome surface uses only first-order terms in the ellipticity parameter ε = a/b - 1 = 0.23, specifically Eqs. (A42) and (A47). This value is not small, and the expansion in Eq. (A41) contains terms proportional to ε^2/(r-a), which are singular as r tends to the inner boundary. The optimal trajectory is reported to hug the inside lane, so the first-order approximation may be used precisely where it is least reliable. The authors should quantify the truncation error in the optimal path and time, for example by comparing with a full-ellipse formulation or by including second-order terms.
[Section 6b, Section 5] Fatigue is introduced only in Section 6b as a scaling argument for the initial conditions and the sprint time, Eq. (6.2). The optimal-control model that produces Fig. 5, however, uses the non-fatiguing pedalling force (5.1), with no fatigue term in the dynamics. Since the final 200 m takes about 9.7 s, longer than the stated ~5 s validity of pure anaerobic respiration, the absence of fatigue in the optimization may materially affect the predicted speed and power over the interval compared with athlete data. The authors should either include fatigue in the optimal-control dynamics or justify its neglect over this time scale before claiming full validation.

minor comments (4)

[Section 6a] The phrase 'an an asymptotic expansion' contains a duplicated article and should read 'an asymptotic expansion'.
[Figure 5 caption] The shaded straight regions in Fig. 5(b,c) are not defined in the caption; please state explicitly which intervals correspond to the straight sections of the track.
[Section 5] The sentence 'In Fig. 2 we plot solutions...' appears after the physiological model is introduced, but the reader may expect the figure to be introduced near the earlier analytical discussion; consider renumbering or moving the figure reference to improve readability.
[Section 6b] The derivation of the initial speed from a balance between constant aerobic power and drag is plausible, but the value 58 km/hr is presented without showing the numerical solution of P0/v = 1/2 ρ CdA v^2; including the intermediate calculation would make the result easier to check.

Circularity Check

1 steps flagged · score 2.0 of 10

Starting-height 'optimum' restates an observed input, but the main trajectory validation is independent.

fitted input called prediction [Section 6(a) Validation setup and Section 6(b) Discussion of parameters and fatigue]
"For this example, we use the same parameter values as before, except with initial positions(0) = 0,n (0) =W cosα(0), and an initial velocity of 58 km/hr purely in the x-direction, which corresponds to the typical speed before descent for Olympic athletes [...] We find that the optimum value of n(0) is always given by n(0) = W cosα(s(0)), which is in accordance with our observation that track cyclists always begin their descent from the top of the velodrome ramp."

The stated 'optimum' n(0) formula is exactly the value imposed as the initial condition for the validation run: the paper sets n(0)=W cosα(0) because that is the observed top-of-ramp starting position, and then reports 'we find' the same top-of-ramp expression as optimal. Since the feasible set is 0 ≤ n ≤ W cosα and n(0) enters the dynamics only through the initial potential energy, the maximizing value is the upper bound already used as input. This is a restatement of the imposed initial condition rather than an independent prediction. It does not reduce the central claim, because the optimal trajectory and the speed/power profiles are still computed from the equations of motion rather than fitted to the measured athlete curves.

full rationale

The main derivation is self-contained against the athlete data: the equations follow from Lagrangian/Newtonian mechanics; the physiological parameters Tmax, omega_max, CdA, D, P0 and tau come from independent published sources; and the initial speed of 58 km/hr is derived from a power balance rather than tuned to the measured power/speed curves. No parameter is fitted to the elite cyclist data used for validation. The computed strategy of descending before the first corner and then hugging the inside lane is a genuine output of the optimal-control problem, not an imposed constraint, and the final-200 m time near 9.7 s is an output of the dynamics. The unbounded lateral-force control noted in Appendix B is a physical-realism or feasibility concern, not a circularity. The only mild circularity is the initial-height discussion, where an observed starting condition is presented as an optimum; this does not bear on the central near-optimality validation.

Assumptions & free parameters 7 free parameters · 8 assumptions · 0 invented entities

The central claim depends on a chain of external inputs: physiological sprint parameters, aerodynamic drag area, fatigue decay constants, and the measured track geometry. The paper also makes modeling simplifications that are not derived, most notably the point-mass rider, the unbounded lateral force, and the first-order ellipticity approximation. None of these inputs are fitted to the athlete data shown in Fig 5, which is why the circularity burden remains low, but they all affect the quantitative predictions and should be audited before use.

free parameters (7)

T_max = 260 Nm (from Dorel et al. [11])
Maximum pedalling torque at zero cadence; sets the pedal force law (5.1) and all velocity/power predictions. Taken from literature, not fitted here, but load-bearing.
omega_max = 25 rad/s (from Dorel et al. [11])
Maximum pedalling cadence; with T_max defines the linear torque-cadence relation.
D = 8.5 m
Gear development chosen as typical for sprint qualification; shifts the equilibrium velocity (5.2) and the power profile.
CdA = 0.22 m^2
Drag area from cycling aerodynamics literature; enters the drag force and the equilibrium/fatigue scalings.
m = 86 kg
Combined cyclist-bike mass, chosen as realistic; scales all forces and the 58 km/hr initial-speed balance.
Fatigue parameters P0/m, Pmax/m, tau = 6.35 W/kg, 17.4 W/kg, 38 s (from Sanders and Heijboer [25])
Fatigue model (6.1); used to derive the 58 km/hr initial speed and the sprint timescale (6.2) that explains s(0).
Initial conditions = s(0)=0, n(0)=W cos alpha(0), v(0)=58 km/hr
Initial conditions imposed in the main velodrome simulation; n(0) and v(0) are justified by observation and energy balance, but s(0)=0 is a hand-chosen example rather than part of the numerical optimization.

assumptions (8)

standard math The Euler-Lagrange equations give the global minimum of the time functionals (2.3) and (3.2) under the stated boundary conditions.
Used in Sections 2 and 3 without proof; standard variational calculus.
domain assumption A cyclist can be modeled as a point mass, with no rolling resistance, bike lean inertia, or steering dynamics.
Adopted in Section 4 and Appendices A-B; needed to reduce the problem to two state variables.
domain assumption Pedal torque is a linearly decreasing function of cadence (5.1), with T_max and omega_max from Dorel et al. [11].
Closes the active-particle dynamics; the paper notes it applies only to pure anaerobic sprinting.
domain assumption Air drag is Fd = 1/2 rho CdA v^2 with constant CdA and no wind or drafting.
Used in Section 4 and the fatigue balance; ignores race interactions.
domain assumption The velodrome surface is exactly z = n tan alpha(s) with constant width W and measured alpha(s).
Surface reconstruction in Section 6(a) and Fig 4.
ad hoc to paper Elliptical curved sections can be represented to first order in epsilon = a/b - 1 = 0.23.
Appendix A(c) expands the ellipse for epsilon much less than 1, which is not satisfied; no error estimate is given.
domain assumption The lateral control F_perp is unbounded in the optimal control problem.
Appendix B treats F_perp as a free control with no grip or normal-reaction constraint.
ad hoc to paper The optimal strategy makes peak velocity occur exactly at the finish line, so sprint time is found by equating fatigue-limited power to drag power there.
Used to derive Eq (6.2) and the s(0) explanation; plausible but not proven by the numerical optimization.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Brachistochrone on a Velodrome." pith.science (2026). https://pith.science/paper/LM4TGFJN

@misc{pith2026190802224,
  author       = {Pith},
  title        = {Pith review of: Brachistochrone on a Velodrome},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LM4TGFJN}},
  note         = {Machine review of arXiv:1908.02224}
}

read the original abstract

The Brachistochrone problem, which describes the curve that carries a particle under gravity in a vertical plane from one height to another in the shortest 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 the velodrome in Montigny le Bretonneux, France. Finally, we discuss the parameters of the problem and the effects of fatigue.

Figures

Figures reproduced from arXiv: 1908.02224 by the authors.

**Figure 1.** (a,b,c,d) Snapshots at different times during a descent trajectory in a velodrome qualification time trial (b,c,d correspond to 0.42, 0.84, 1.28 s after the descent). (e) Top five track times taken from the qualification round of the Rio 2016 Olympic games [10]. 1. 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 ind… view at source ↗

**Figure 2.** Cycling on the planar Brachistochrone (a) and the Brachisto-‘cone’ (b). In the planar case, we choose a slope angle of α = 20◦ and a total distance of L = 40 m. In the cone case we take α = 10◦, a starting radius of R0 cos α = 10 m and a total rotation of θ1 − θ0 = π/4. Owing to the more complicated form of (3.3) neither an explicit nor a parameterised closed form solution is available (though one constant of integr… view at source ↗

**Figure 3.** Study of the cycling physiology parameters. In the base case, we have Tmax = 260 Nm, ωmax = 25 rad/s, and D = 8.5 m. In each of the other cases, we vary just one of these parameters and plot the trajectory (a) and the velocity (c). The equilibrium velocity (5.2) for each case is indicated in (c), and the dependence on D is shown in the insert (b). 5. Cyclist physiology To model the pedalling force Fp(t), there are c… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: (a) Three-dimensional reconstruction of the velodrome of Montigny le Bretonneux. (b) The velodrome slope angle α measured experimentally as a function of distance around the track s. (c) Schematic diagram of the velodrome surface z = n tan α(s), illustrating the tangen…

**Figure 5.** Figure 5: (a) Brachistochrone on a velodrome, viewed from above, illustrating the positions that correspond to the snapshots in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Schematic diagram showing how we calculate the distance d(φ) between a point in the plane (r,θ) and a point on the ellipse (R(φ),φ). or written in terms of the rotated variable R = r/ cos α, we get d dt 1 2 mv 2 + mgR sin α = vFk . (A 31) (c) Calculating surface gr…

**Figure 7.** Figure 7: Optimum descent trajectories for different starting positions between s(0) = 7 m and s(0) = −20 m. T = Nδt. The governing differential equations are discretised using a second order forward Euler scheme, producing a system of algebraic equations fj (X, F ) = 0, j = 1 .…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

[1]

16rspa.royalsocietypublishing.org Proc R Soc A 0000000

LN Hand and JD Finch. 16rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical mechanics. Cambridge University Press, 1998

work page 1998
[2]

The brachistochrone problem and modern control theory

HJ Sussmann and JC Willems. The brachistochrone problem and modern control theory. Contemporary trends in nonlinear geometric control theory and its applications (Mexico City, 2000) , pages 113–166, 2002

work page 2000
[3]

A History of the Calculus of Variations from the 17th through the 19th Century , volume 5

HH Goldstine. A History of the Calculus of Variations from the 17th through the 19th Century , volume 5. Springer Science & Business Media, 2012

work page 2012
[4]

Brachistochrone with coulomb friction

N Ashby, WE Brittin, WF Love, and W Wyss. Brachistochrone with coulomb friction. American Journal of Physics , 43(10):902–906, 1975

work page 1975
[5]

On the analytical solution of the brachistochrone problem in a non-conservative ﬁeld

B Vratanar and M Saje. On the analytical solution of the brachistochrone problem in a non-conservative ﬁeld. International Journal of Non-Linear Mechanics , 33(3):489–505, 1998

work page 1998
[6]

Brachistochrone with coulomb friction

JC Hayen. Brachistochrone with coulomb friction. International Journal of Non-Linear Mechanics , 40(8):1057–1075, 2005

work page 2005
[7]

Time-optimal control of rolling bodies

G Perantoni and DJN Limebeer. Time-optimal control of rolling bodies. International Journal of Control , 86(11):2006–2021, 2013

work page 2006
[8]

The elastochrone: the descent time of a sphere on a ﬂexible beam

JM Aristoff, C Clanet, and JWM Bush. The elastochrone: the descent time of a sphere on a ﬂexible beam. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 465(2107):2293–2311, 2009

work page 2009

Show all 29 references

[9]

Time-optimal quantum evolution

A Carlini, A Hosoya, T Koike, and Y Okudaira. Time-optimal quantum evolution. Physical Review Letters, 96(6):060503, 2006

2006
[10]

https://assetrio2016.azureedge.net/_odf-documents/c/t/CTM001900_ Results_2016_08_12_2797f6c2_59ec_47cc_b9ba_934d32474041.pdf, 2019

Rio 2016 archives. https://assetrio2016.azureedge.net/_odf-documents/c/t/CTM001900_ Results_2016_08_12_2797f6c2_59ec_47cc_b9ba_934d32474041.pdf, 2019. Accessed: 23-07-2019

2016
[11]

Torque and power-velocity relationships in cycling: relevance to track sprint performance in world-class cyclists

S Dorel, CA Hautier, O Rambaud, David Rouffet, E Van Praagh, J-R Lacour, and M Bourdin. Torque and power-velocity relationships in cycling: relevance to track sprint performance in world-class cyclists. International Journal of Sports Medicine , 26(09):739–746, 2005

2005
[12]

The science of cycling

EW Faria, DL Parker, and IE Faria. The science of cycling. Sports Medicine, 35(4):285–312, 2005

2005
[13]

Mathematical physics

E Butkov. Mathematical physics. Addison Wesley Publishing Co, 1968

1968
[14]

Mechanics of cycling

JP Mariot. Mechanics of cycling. Revue de Physique Appliquée , 19(4):349–357, 1984

1984
[15]

Aerodynamic drag and biomechanical power of a track cyclist as a function of shoulder and torso angles

L Underwood, J Schumacher, J Burette-Pommay, and M Jermy. Aerodynamic drag and biomechanical power of a track cyclist as a function of shoulder and torso angles. Sports Engineering, 14(2-4):147–154, 2011

2011
[16]

Flow topology in the wake of a cyclist and its effect on aerodynamic drag

TN Crouch, D Burton, NAT Brown, MC Thompson, and J Sheridan. Flow topology in the wake of a cyclist and its effect on aerodynamic drag. Journal of Fluid Mechanics , 748:5–35, 2014

2014
[17]

Riding against the wind: a review of competition cycling aerodynamics

TN Crouch, D Burton, ZA LaBry, and KB Blair. Riding against the wind: a review of competition cycling aerodynamics. Sports Engineering, 20(2):81–110, 2017. 17rspa.royalsocietypublishing.org Proc R Soc A 0000000

2017
[18]

The heat of shortening and the dynamic constants of muscle

AV Hill. The heat of shortening and the dynamic constants of muscle. Proceedings of the Royal Society of London. Series B-Biological Sciences , 126(843):136–195, 1938

1938
[19]

Numerical optimization, second edition

J Nocedal and SJ Wright. Numerical optimization, second edition . Springer, 2006

2006
[20]

On the implementation of an interior-point ﬁlter line-search algorithm for large-scale nonlinear programming

A Wächter and LT Biegler. On the implementation of an interior-point ﬁlter line-search algorithm for large-scale nonlinear programming. Mathematical Programming, 106(1):25–57, 2006

2006
[21]

Validation of a mathematical model for road cycling power

JC Martin, DL Milliken, JE Cobb, KL McFadden, and AR Coggan. Validation of a mathematical model for road cycling power. Journal of Applied Biomechanics , 14(3):276–291, 1998

1998
[22]

The understanding and development of cycling aerodynamics

RA Lukes, SB Chin, and SJ Haake. The understanding and development of cycling aerodynamics. Sports Engineering, 8(2):59–74, 2005

2005
[23]

An experimental study on aerodynamic performance of time trial bicycle helmets

H Chowdhury and F Alam. An experimental study on aerodynamic performance of time trial bicycle helmets. Sports Engineering, 17(3):165–170, 2014

2014
[24]

Understanding sprint-cycling performance: the integration of muscle power, resistance, and modeling

JC Martin, CJ Davidson, and ER Pardyjak. Understanding sprint-cycling performance: the integration of muscle power, resistance, and modeling. International Journal of Sports Physiology and Performance , 2(1):5–21, 2007

2007
[25]

The anaerobic power reserve and its applicability in professional road cycling

D Sanders and M Heijboer. The anaerobic power reserve and its applicability in professional road cycling. Journal of Sports Sciences , 37(6):621–629, 2019

2019
[26]

Cyclist drag in team pursuit: inﬂuence of cyclist sequence, stature, and arm spacing

T Defraeye, B Blocken, E Koninckx, P Hespel, P Verboven, B Nicolai, and J Carmeliet. Cyclist drag in team pursuit: inﬂuence of cyclist sequence, stature, and arm spacing. Journal of Biomechanical Engineering , 136(1):011005, 2014

2014
[27]

Aerodynamics of human-powered vehicles

CR Kyle and MD Weaver. Aerodynamics of human-powered vehicles. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy , 218(3):141–154, 2004

2004
[28]

Jump: A modeling language for mathematical optimization

I Dunning, J Huchette, and M Lubin. Jump: A modeling language for mathematical optimization. SIAM Review, 59(2):295–320, 2017

2017
[29]

Julia: A fresh approach to numerical computing

J Bezanson, A Edelman, S Karpinski, and VB Shah. Julia: A fresh approach to numerical computing. SIAM Review, 59(1):65–98, 2017

2017

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

16rspa.royalsocietypublishing.org Proc R Soc A 0000000

LN Hand and JD Finch. 16rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical mechanics. Cambridge University Press, 1998

work page 1998

[2] [2]

The brachistochrone problem and modern control theory

HJ Sussmann and JC Willems. The brachistochrone problem and modern control theory. Contemporary trends in nonlinear geometric control theory and its applications (Mexico City, 2000) , pages 113–166, 2002

work page 2000

[3] [3]

A History of the Calculus of Variations from the 17th through the 19th Century , volume 5

HH Goldstine. A History of the Calculus of Variations from the 17th through the 19th Century , volume 5. Springer Science & Business Media, 2012

work page 2012

[4] [4]

Brachistochrone with coulomb friction

N Ashby, WE Brittin, WF Love, and W Wyss. Brachistochrone with coulomb friction. American Journal of Physics , 43(10):902–906, 1975

work page 1975

[5] [5]

On the analytical solution of the brachistochrone problem in a non-conservative ﬁeld

B Vratanar and M Saje. On the analytical solution of the brachistochrone problem in a non-conservative ﬁeld. International Journal of Non-Linear Mechanics , 33(3):489–505, 1998

work page 1998

[6] [6]

Brachistochrone with coulomb friction

JC Hayen. Brachistochrone with coulomb friction. International Journal of Non-Linear Mechanics , 40(8):1057–1075, 2005

work page 2005

[7] [7]

Time-optimal control of rolling bodies

G Perantoni and DJN Limebeer. Time-optimal control of rolling bodies. International Journal of Control , 86(11):2006–2021, 2013

work page 2006

[8] [8]

The elastochrone: the descent time of a sphere on a ﬂexible beam

JM Aristoff, C Clanet, and JWM Bush. The elastochrone: the descent time of a sphere on a ﬂexible beam. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 465(2107):2293–2311, 2009

work page 2009