On the brachistochrone problem for cycling ascents

Len Bos; Michael A. Slawinski; Rapha\"el A. Slawinski; Theodore Stanoev

arxiv: 2511.13509 · v5 · submitted 2025-11-17 · ⚛️ physics.pop-ph

On the brachistochrone problem for cycling ascents

Len Bos , Michael A. Slawinski , Rapha\"el A. Slawinski , Theodore Stanoev This is my paper

Pith reviewed 2026-05-17 20:52 UTC · model grok-4.3

classification ⚛️ physics.pop-ph

keywords brachistochronecycling ascentVAMpower constraintoptimal trajectorystraight-line pathconstant power

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The pith

Under fixed average power, the fastest ascent between two points is a straight line at constant speed.

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

The paper investigates the brachistochrone problem for cyclists climbing under a fixed average-power limit. It establishes that, to reach a given height gain in minimum time, the optimal trajectory is the straight line between start and end points ridden at constant speed. This constant speed corresponds to constant power along that line. The finding contrasts with the classical descent brachistochrone, which follows a cycloid with varying speed. The result also supports choosing the steepest feasible constant-grade climb to maximize VAM, subject to practical limits such as cadence and balance.

Core claim

For given start and end points and any fixed average-power constraint, the brachistochrone for minimum ascent time is the straight line connecting the points, covered at constant speed; along such a line this speed profile is equivalent to constant power. This holds in contrast to the classical gravity-driven brachistochrone, which is a cycloid along which speed varies.

What carries the argument

The power-constrained brachistochrone: the path that minimizes time to a fixed height gain when average power output is held constant, which reduces to straight-line constant-speed motion.

If this is right

For any fixed power budget the minimal-time path is always the direct straight line at steady speed.
VAM is therefore maximized by selecting the steepest feasible constant-grade straight climb rather than varying the trajectory.
Constant power output along the straight path is optimal and equivalent to the required constant speed.
Practical limits on steepness arise from pedaling cadence, balance, and wheel traction rather than from path curvature.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Route planners or race strategists could prioritize elevation profiles that allow sustained high power on direct lines rather than seeking curved or variable-grade alternatives.
Training emphasis might shift toward sustaining high steady power output instead of optimizing for changing gradients or cornering.
Extensions to models that include rolling resistance or wind could reveal whether the straight-line result survives once path-dependent losses are added.

Load-bearing premise

Power output converts directly into motion against gravity with no dissipative forces or speed-dependent losses that would make the time depend on the specific path shape.

What would settle it

Measure ascent times for a cyclist maintaining identical average power while following a straight-line path versus a curved path between the same two points; if the straight path is not faster, the claim is refuted.

Figures

Figures reproduced from arXiv: 2511.13509 by Len Bos, Michael A. Slawinski, Rapha\"el A. Slawinski, Theodore Stanoev.

**Figure 1.** Figure 1: V (θ): Optimal constant ground speed as a function of slope The vertical component of that speed is V (θ) sin θ, which is shown in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: V (θ) sin θ: Vertical speed as a function of slope The ascent time for the elevation gain of 1 000 m is shown in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: T(θ): Ascent time for the elevation gain of 1 000 m as a function of slope 4 Discussion and conclusion As stated by Lemmas 1 and 2, respectively, the optimal ascent to maximize the VAM — for a given average power — is a constant-grade hill with the highest grade that a cyclist can handle. Since the most efficient ascent is achieved with a constant speed, it follows that, for a constant grade, it is equival… view at source ↗

read the original abstract

VAM ({\it velocit\`a ascensionale media}) is a measurement that quantifies a cyclist's climbing ability. We show that to minimize the time to attain a given height gain\, -- \,which is tantamount to maximizing VAM\, -- \,a cyclist should climb as steep a constant-grade hill as possible. Apart from the power-to-weight ratio, the limit of steepness is imposed by such factors as the efficiency of pedalling, which is related to feasible cadence, maintaining balance, preventing lifting of the front, and skidding of the rear, wheel. In an appendix, we discuss steepness constraints due to pedalling efficiency. The article itself is focused on consequences of the power available to the cyclist, which can be viewed as a necessary condition to examine other aspects of climbing strategy. We show that\, -- \,for given start and end points, and for any fixed average-power constraint\, -- \,the brachistochrone, which is the trajectory of minimum ascent time, is the straight line connecting these points, covered with a constant speed, which along such a line is equivalent to a constant power. This is in contrast to the classical solution of a descent brachistochrone under gravity, which is a cycloid along which the speed is not constant.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims straight-line constant-speed ascents minimize time under fixed average power for cycling climbs, but this uniqueness looks fragile once dissipation is absent.

read the letter

The main thing to know is that the authors set up the brachistochrone for powered ascent under a fixed average-power limit and conclude the minimum-time path is the straight line at constant speed. This is presented as a direct consequence of the average-power constraint and the start/end points, and they contrast it explicitly with the classical cycloid for gravity-driven descents. They also flag that power is only a necessary condition and push other limits like cadence and balance into an appendix. That separation is useful and keeps the scope clean. The writing is straightforward and the central claim is stated without much hedging. What the paper does well is to make the power-to-VAM link explicit and to note that constant speed on the straight line turns average power into constant power. The math appears to rest on a variational setup that treats the average-power constraint as the objective driver. On the soft spots, the uniqueness of the straight line is the part that needs checking. The stress-test observation holds weight here: without dissipative terms, the work-energy relation gives total energy input equal to mgH plus net kinetic-energy change. Any connecting curve can in principle reach the same minimum time by a speed profile that keeps net Delta KE at zero while delivering exactly mgH of work. The paper would need to show why other paths cannot satisfy the average-power constraint under that condition, or whether an implicit assumption rules out speed variation. No numerical checks or explicit functional appear in the material I have, so it is hard to judge how load-bearing that step is. The citation pattern is light, which fits a short note. This is for people who already work on variational problems in sports or who model climbing performance. A reader who knows the classical brachistochrone will get the contrast quickly and can test the power-constraint setup themselves. It is coherent enough on its own terms to deserve a serious referee who can verify the derivation and ask whether the model needs an explicit loss term to make the straight line unique.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that, for a cyclist ascending between fixed start and end points under any fixed average-power constraint, the brachistochrone (minimum-time trajectory) is the straight line connecting the points, traversed at constant speed (which corresponds to constant power along that line). This is presented as a contrast to the classical cycloidal brachistochrone for unpowered descents under gravity. The paper also discusses VAM maximization and, in an appendix, practical steepness limits arising from pedaling efficiency, balance, and wheel traction.

Significance. If the central claim were correct, the result would imply a simple geometric optimum for climbing strategy: cyclists should select the steepest feasible constant-grade straight-line ascent and maintain constant speed to maximize VAM under a given power budget. This would offer a straightforward, parameter-free prediction for route planning and training, distinct from the curved optima in passive gravitational problems.

major comments (2)

[Abstract] Abstract and main claim: the assertion that the straight line at constant speed is the unique brachistochrone under a fixed average-power constraint is not supported. The work-energy theorem gives total energy input E = mgH + ΔKE; with average power P_avg fixed, time T = E/P_avg. The minimum T = mgH/P_avg is therefore achieved on any path for which a speed profile exists that begins and ends at the same speed (ΔKE = 0) while the cyclist supplies exactly mgH. No further path dependence appears in the absence of dissipation, so the minimal time is not unique to the straight line.
[Main text] Main text (derivation section): the functional to be minimized is never stated explicitly, nor is the variational problem formulated. It is therefore impossible to verify how the average-power constraint is imposed, whether power is taken as constant or only its time average is fixed, or why any deviation from the straight line necessarily increases T.

minor comments (1)

[Appendix] The appendix on pedaling-efficiency limits would be strengthened by quantitative references to cadence-efficiency curves or biomechanical data rather than qualitative discussion alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment below and note the revisions that will be incorporated to improve the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract and main claim: the assertion that the straight line at constant speed is the unique brachistochrone under a fixed average-power constraint is not supported. The work-energy theorem gives total energy input E = mgH + ΔKE; with average power P_avg fixed, time T = E/P_avg. The minimum T = mgH/P_avg is therefore achieved on any path for which a speed profile exists that begins and ends at the same speed (ΔKE = 0) while the cyclist supplies exactly mgH. No further path dependence appears in the absence of dissipation, so the minimal time is not unique to the straight line.

Authors: We agree with the referee's application of the work-energy theorem. In the absence of dissipation, any trajectory permitting a speed profile with ΔKE = 0 requires the cyclist to supply exactly mgH of energy, yielding the same minimal time T = mgH/P_avg for a fixed average power. The straight line at constant speed is one such trajectory (where constant speed implies constant power), but it is not unique. Our original emphasis was on the practical benefit of constant power output for rider consistency. We will revise the abstract and related claims to state that minimal time is achieved by any qualifying path, while highlighting the straight line as the case of constant power. revision: yes
Referee: [Main text] Main text (derivation section): the functional to be minimized is never stated explicitly, nor is the variational problem formulated. It is therefore impossible to verify how the average-power constraint is imposed, whether power is taken as constant or only its time average is fixed, or why any deviation from the straight line necessarily increases T.

Authors: We acknowledge that the manuscript does not explicitly present the objective functional or the full variational setup. In the revision we will state the problem as minimizing ascent time T subject to the integral constraint (1/T) ∫ P(t) dt = P_avg, with P(t) = F_prop(t) · v(t) and the equations of motion including gravity. We will show that, without dissipation, T reduces to mgH/P_avg independently of path (provided ΔKE = 0), and clarify that the straight-line constant-speed case is singled out because it corresponds to constant instantaneous power rather than because it uniquely minimizes T. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives its central claim directly from the fixed average-power constraint together with the work-energy relation applied to given start and end points. The conclusion that the straight-line constant-speed trajectory minimizes ascent time is presented as following from these inputs under the model's assumptions of no dissipation and Delta KE = 0. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the derivation remains self-contained against the stated physical premises even if the uniqueness claim admits independent physical debate.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the assumption that power can be applied to balance gravity exactly at constant speed on a constant grade and that no other forces make time path-dependent.

axioms (1)

domain assumption Power supplied by the cyclist can be converted directly into motion against the gravitational component along the path with no dissipative losses.
This assumption is required for the constant-speed constant-power equivalence on the straight line and for the claim that the straight line is uniquely optimal.

pith-pipeline@v0.9.0 · 5540 in / 1244 out tokens · 112319 ms · 2026-05-17T20:52:15.890425+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

Benham, G., Cohen, C., Brunet, E., and Clanet, C. (2020). Brachistochrone on a velodrome.Proceedings of the Royal Society A, https://doi.org/10.1098/rspa.2020.0153

work page doi:10.1098/rspa.2020.0153 2020
[2]

A., Slawinski, R

Bos, L., Slawinski, M. A., Slawinski, R. A., and Stanoev, T. (2024). On minimizing cyclists’ ascent times. Dolomites Research Notes on Approximation, 17(3):5–19

work page 2024
[3]

A., Slawinski, R

Bos, L., Slawinski, M. A., Slawinski, R. A., and Stanoev, T. (2025a). On minimizing cyclists’ ascent times: Part I.arXiv, 2403.03363 [physics.class-ph]. 6

work page arXiv
[4]

A., Slawinski, R

Bos, L., Slawinski, M. A., Slawinski, R. A., and Stanoev, T. (2025b). On minimizing cyclists’ ascent times: Part II.arXiv, arXiv:2503.03235 [physics.class-ph]

work page arXiv
[5]

A., Slawinski, R

Bos, L., Slawinski, M. A., Slawinski, R. A., and Stanoev, T. (2025c). On minimizing cyclists’ ascent times: Part II.Mathematics and Mechanics of Solids, https://doi.org/10.1177/10812865251343512

work page doi:10.1177/10812865251343512
[6]

A., and Stanoev, T

Bos, L., Slawinski, M. A., and Stanoev, T. (2021). On maximizing VAM for a given power: Slope, cadence, force and gear-ratio considerations.arXiv, 2006.15816 [physics.class-ph]. 7

work page arXiv 2021

[1] [1]

Benham, G., Cohen, C., Brunet, E., and Clanet, C. (2020). Brachistochrone on a velodrome.Proceedings of the Royal Society A, https://doi.org/10.1098/rspa.2020.0153

work page doi:10.1098/rspa.2020.0153 2020

[2] [2]

A., Slawinski, R

Bos, L., Slawinski, M. A., Slawinski, R. A., and Stanoev, T. (2024). On minimizing cyclists’ ascent times. Dolomites Research Notes on Approximation, 17(3):5–19

work page 2024

[3] [3]

A., Slawinski, R

Bos, L., Slawinski, M. A., Slawinski, R. A., and Stanoev, T. (2025a). On minimizing cyclists’ ascent times: Part I.arXiv, 2403.03363 [physics.class-ph]. 6

work page arXiv

[4] [4]

A., Slawinski, R

Bos, L., Slawinski, M. A., Slawinski, R. A., and Stanoev, T. (2025b). On minimizing cyclists’ ascent times: Part II.arXiv, arXiv:2503.03235 [physics.class-ph]

work page arXiv

[5] [5]

A., Slawinski, R

Bos, L., Slawinski, M. A., Slawinski, R. A., and Stanoev, T. (2025c). On minimizing cyclists’ ascent times: Part II.Mathematics and Mechanics of Solids, https://doi.org/10.1177/10812865251343512

work page doi:10.1177/10812865251343512

[6] [6]

A., and Stanoev, T

Bos, L., Slawinski, M. A., and Stanoev, T. (2021). On maximizing VAM for a given power: Slope, cadence, force and gear-ratio considerations.arXiv, 2006.15816 [physics.class-ph]. 7

work page arXiv 2021