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REVIEW 4 major objections 4 minor 22 references

Optimal Snake Locomotion on Flat Surfaces: An Analytical Framework

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For snakes on flat ground, the optimal body shape for both efficiency and speed is a sawtooth, with the tooth angle set by the friction ratio.

desk verdict Useful closed-form formulas for sawtooth snake gaits, but the central variational proof treats U—a functional of θ(s)—as a constant, so the sawtooth-optimality claim does not follow as written. read the letter →

arxiv 2412.03725 v1 pith:APTMAQ7G submitted 2024-12-04 physics.bio-ph

Yogesh Phalak , Hodjat Pendar This is my paper · ORCID
classification physics.bio-ph
keywords snakelocomotionfrictionanisotropyresistiveforcetheoryvariationaloptimizationcostoftransportsawtoothwaveformundulatorylowReynoldsnumber
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

The paper derives an analytical answer to an old question: how should a snake shape its body to slither efficiently, or as fast as possible, on flat ground? Treating the snake as a uniform, flexible, rigid-free body whose interaction with the ground is anisotropic Coulomb friction, it writes both the cost of transport and the forward speed as integrals over the body-shape function and applies the Euler-Lagrange equations. The conclusion is that the optimal body shape is a sawtooth, θ(s)=±constant, with the tooth angle fixed by the friction ratio μ=μ_n/μ_f through closed-form expressions. If correct, the same sawtooth conclusion holds for any slender-body propulsion governed by resistive force theory, including sand-swimming and viscous-fluid swimming.

What carries the argument

The central object is the body shape function θ(s), the local tangent angle relative to the direction of travel, defined over one period of the undulation. The paper expresses both the cost of transport (Eq. 12) and the average speed (Eq. 9) as functionals of θ(s) with no θ′ dependence, so the Euler-Lagrange equations collapse to algebraic stationarity conditions rather than differential equations. For efficiency the condition is that cosψ, the component of local velocity tangent to the body, be constant; for speed it is that cos(θ−φ) be constant. The force-balance equations then select θ=±constant, i.e. a sawtooth, and the closed-form relations (Eqs. 14, 16, 19) connect the sawtooth angle and wavelength to the friction ratio μ.

What would settle it

Pick a friction ratio such as μ=1.5, solve the full force-balance equations (Eqs. 8–10) exactly for a family of smooth sinusoidal body shapes, and compare their cost of transport and speed with the sawtooth values from Eqs. 16 and 19; if any non-sawtooth shape beats the sawtooth on either objective, the variational reduction is incorrect. The same check can be done experimentally with a snake robot whose ventral friction is tuned to μ=1.5.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the efficiency-optimal and speed-optimal gaits coincide in form: both are piecewise-linear sawtooth waves with θ(s)=±constant, not smooth sinusoidal curves. The argument starts from the cost of transport $η^{{-1}}$ = (1/U)∫ cosψ ds and the speed U = ∫ v cos(θ+ψ−φ) ds (Eqs. 12 and 9), assumes the average speed U can be held fixed during shape variation, and reduces the Euler-Lagrange condition to a pointwise constancy condition on cosψ (efficiency) or on cos(θ−φ) (speed). Combined with the transverse force balance, this forces θ to be constant in magnitude. The paper then derives U = λ − λ/(λ²+μ(1−λ²)) (Eq. 14), a sixth-order polynomial in λ for the efficiency optimum (Eq. 16), and the closed-form speed-optimal wavelength λ = $\sqrt$((2μ+1 − $\sqrt$(8μ+1))/(2(μ−1))) (Eq. 19). The same variational step applied to granular and viscous resistive-force laws yields the same sawtooth optimum, suggesting the result is a general feature of RFT-based undulatory locomotion.

Load-bearing premise

The derivation treats the average forward speed U as a fixed number when the body shape is varied, even though U is itself an integral over that shape (Eq. 9); if a small shape change changes U, the Euler-Lagrange step forcing cosψ to be constant does not follow.

Editorial extensions

If this is right

  • An optimally efficient snake on a flat surface should adopt a sawtooth waveform; for the experimentally observed range 1<μ<2, the optimal efficiency angle stays near 45°, while the speed-optimal angle is steeper at low μ.
  • The speed-optimal wavelength has a closed form, λ = sqrt((2μ+1 − sqrt(8μ+1))/(2(μ−1))), and its wave amplitude scales as A ∼ (1/2)(μ/2)^(−1/4) at large μ, matching the asymptotic behavior of prior numerical optimization.
  • Because the same stationarity condition arises for granular and viscous resistive-force laws, the sawtooth conclusion extends to sand-swimming lizards and flagellar swimming, not just snake slithering.
  • The analytical results provide a parameter-free upper bound for efficiency and speed in more realistic finite-length snakes, where body rigidity and unsteady average speed add dissipation.

Reading between the lines

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

  • If the sawtooth claim survives scrutiny, it implies that the smooth, nearly sinusoidal body waves observed in many real snakes are not optimal in the friction-only model; the discrepancy is a plausible signature of body stiffness, actuation constraints, or finite-length effects that the model deliberately omits.
  • A testable extension: measure the local tangent angle of an actual snake or snake robot on surfaces with engineered friction ratios, and compare the distribution of θ to the predicted sawtooth angle; a clear concentration near the predicted angle would support the model, while broad smooth distributions would indicate that omitted costs matter.
  • The variational reduction to a pointwise constancy condition may generalize to other RFT settings, e.g., swimming near a wall or in a stratified fluid, where the force law changes the stationarity condition and possibly selects different optimal shapes.
  • Because the paper treats U as fixed in the variation, a corrected variational treatment that accounts for δU could either confirm the sawtooth or shift the optimum; re-deriving Eq. 16 with the full functional derivative is a direct mathematical check.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper develops an analytical variational framework for optimal locomotion of a snake modeled as an inextensible, flexible, periodic body moving on a flat surface under Coulomb friction with anisotropic coefficients. It claims that minimizing the cost of transport and maximizing forward speed both lead to sawtooth body shapes, i.e., θ(s)=±constant, with the optimal angle determined by the friction ratio μ. The paper derives closed-form expressions for the optimal speed and efficiency as functions of wavelength λ (Eqs. 14–16, 19), compares the results with Alben's numerical optimization, and extends the same variational argument to sand-swimming and viscous-fluid propulsion. The central mathematical step in Section 3 is a Euler–Lagrange treatment of integrands that contain no derivative of θ, leading to the conclusion that cos ψ and then cos θ must be constant.

Significance. If the main derivation were correct, the paper would provide a compact analytical explanation for a range of numerical observations, including the emergence of sawtooth-like optimal shapes at low transverse friction, and would unify Lighthill's flagellar result with snake locomotion and other RFT media. The manuscript has clear strengths: it presents an explicit non-dimensional model, gives closed-form expressions for the restricted sawtooth family, reports no fitted parameters, and includes a direct comparison with Alben's independent numerical optimization. However, the purported variational proof of global optimality is invalid as written, because it treats the average speed and other global kinematic quantities as constants when they are functionals of the shape. Consequently, the central claim that the optimal shape must be a sawtooth is not established by the derivation, and the paper's main contribution reduces to an analysis of a one-parameter family of candidate shapes.

major comments (4)
  1. [§3.1, Eqs. (12)–(13)] The derivation of the optimal efficiency condition is not a valid variational step. Equation (12) defines η^{-1} = (1/U) ∫ cos ψ ds, and Eq. (9) defines U as an integral over v(s), which itself depends on θ(s) through Eqs. (1)–(3) and the equilibrium equations. When the shape is varied, δU is generally nonzero, so the first variation contains the term −(η^{-1}/U) δU. The paper sets ∂/∂θ(cos ψ/U)=0, which treats U as a constant independent of θ. This is explicitly contradicted by the text immediately before Eq. (13), which states that all kinematic variables, including U, are functions of θ(s), and that the explicit form of U in terms of θ(s) is unknown. Since the constancy of cos ψ is the premise for the entire sawtooth conclusion, this is a load-bearing error.
  2. [§3.2, Eqs. (17)–(18)] The same functional-derivative defect appears in the speed maximization. Equation (17) applies ∂/∂θ to the integrand v cos(θ+ψ−ϕ) and concludes that this quantity is constant. But v, ψ, and ϕ are not local functions of θ(s); they are determined globally by the force-balance equations (8)–(10). Equation (18) makes the dependence explicit: the right-hand side contains W and θ0, which are s-independent but shape-dependent. A correct stationarity condition must account for variations of W, θ0, ϕ, and ψ induced by a change in θ(s). The paper does not do this, so the derivation of Eq. (19) is unsupported.
  3. [§3, closure constraint] The admissible class of body shapes is restricted by the continuity/closure condition ∫ sin θ ds=0, as stated in Section 3. However, the Euler–Lagrange equations (13) and (17) are derived without introducing this constraint via a Lagrange multiplier or otherwise restricting the variation space. As a result, the derived necessary conditions apply to a larger set of shapes than the admissible ones. Even if the rest of the variational reasoning were correct, optimality among admissible periodic shapes would not follow without enforcing this constraint.
  4. [§4 and Figure 3] The comparison with Alben's numerical optimization and the asymptotic match in Eq. (20) are not sufficient to validate the optimality claim. Because the sawtooth conclusion is derived from the invalid variational steps, the agreement in Figure 3 only shows that the one-parameter sawtooth family contains shapes with favorable efficiency; it does not demonstrate that these shapes are global optimizers. The statement that the numerical and analytical results are 'closely aligned' therefore overstates what the analytical framework actually proves.
minor comments (4)
  1. [§1 and §2, typos] There are several typographical errors: 'Rwynolds number' should be 'Reynolds number', 'viscus' should be 'viscous', 'Heavyside' should be 'Heaviside', and 'non-dimensionalion' should be 'non-dimensionalization'.
  2. [§3.1 and §3.2, notation] The notation θ=±constant is used interchangeably with 'sawtooth shape,' but a periodic function composed of segments with slopes ±θ is not uniquely a sawtooth; the paper acknowledges this, yet the phrasing 'a sawtooth shape is the most natural shape' is informal. Clarifying the exact class of shapes and how the optimal one is selected within that class would improve precision.
  3. [§4, Figure 3] The caption states that the numerical results are 'reconstructed from [7]' but does not describe the reconstruction procedure or its accuracy. Since this comparison is used to support the agreement claim, more detail is needed.
  4. [References] In Section 1, the text 'David Hu et al. [5] also demonstrated' cites reference [5], which is Lighthill's book, not the experimental study by Hu et al. The intended citation appears to be [1] or [2].

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the sawtooth conclusion is not built into the inputs; the variational gap is a derivation error, not an assumed conclusion, and the paper does not fit parameters to data.

full rationale

The derivation chain is not circular. The paper defines the cost of transport and the average speed as functionals of the body shape θ(s) (Eqs. 9 and 12), and then attempts a variational optimization. The known mathematical issue is that the paper treats quantities such as U, W, and θ0, which are functionals of θ(s), as constants when applying the Euler–Lagrange equation (Eqs. 13 and 17). This is a substantive technical error, but it is not circularity: the conclusion that cos ψ is constant, and hence that θ is piecewise constant, does not coincide with an input assumption or a fitted parameter. The paper explicitly acknowledges it does not know the explicit functional form of U in terms of θ(s), but then proceeds as if U were independent of the variation; this is an unjustified step, not a self-referential definition. No parameter is fitted to the data used for comparison, and the comparison to Alben's numerical optimization is external and independent. The closed-form expressions for U, the efficiency polynomial, and the optimal speed wavelength are derived from the restricted sawtooth family and are not equivalent to the variational inputs by construction. Self-citation is not load-bearing, since no cited result by the present authors is used to justify the central claim. Therefore, the paper should be scored as having no significant circularity, with the mathematical gap noted as a separate correctness concern rather than a circularity finding.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The framework rests on resistively force theory and the Hu-Shelley friction model from prior literature, plus idealized body assumptions. No new physical entities are introduced, and no free parameters are fitted to data. The main unsupported input is the way the variational problem is set up in Section 3.

assumptions (5)
  • domain assumption Friction force on each body segment follows the Hu-Shelley Coulomb model F = -rho g (mu_n sin psi e_n + mu_t cos psi e_t) (Eq. 4).
    Adopted from experimental work [1,2]; it is the core constitutive law on which all subsequent force balance and cost equations rest.
  • domain assumption The snake body is uniform, inextensible, perfectly flexible with no bending stiffness, and moves in a horizontal plane with steady velocity and no rotation over infinitely many periodic cycles.
    Stated in Section 2 and used to justify the time-independent, single-cycle analysis; the authors acknowledge in Section 5 that finite length, stiffness, and rotation are excluded.
  • domain assumption All body points slide in the forward direction, so the Heaviside friction switch in Eq. 4 is never activated to the backward branch.
    Initially assumed in Section 2; the paper asserts a later proof of psi < pi/2 or psi > pi/2, but no rigorous proof is given. The cost and force expressions, Eqs. 7, 11, and 12, depend on this single-branch form.
  • ad hoc to paper The average speed U may be treated as constant when taking the variation of eta^{-1} = (1/U) integral of cos(psi) ds.
    This is the unsupported variational premise after Eq. 13; U is actually a functional of theta(s) through Eq. 9, so the variation should include delta U.
  • ad hoc to paper The constraint integral of sin(theta) ds = 0 can be enforced without a Lagrange multiplier in the Euler-Lagrange equations.
    Mentioned in Section 3 as the only constraint, but the variational derivatives in Eqs. 13 and 17 do not include a multiplier term for it.

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Cite this review

Pith. "Pith review of Optimal Snake Locomotion on Flat Surfaces: An Analytical Framework." pith.science (2026). https://pith.science/paper/APTMAQ7G

@misc{pith2026241203725,
  author       = {Pith},
  title        = {Pith review of: Optimal Snake Locomotion on Flat Surfaces: An Analytical Framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/APTMAQ7G}},
  note         = {Machine review of arXiv:2412.03725}
}
read the original abstract

In this theoretical study, we present an analytical framework to investigate the slithering motion of snakes on flat surfaces. While previous studies have predominantly relied on numerical methods to identify optimal locomotion kinematics, such approaches are often sensitive to initial guesses and the number of kinematic parameters in the model. Here, we derive analytical solutions for optimal kinematics that minimize the cost of transport or maximize the velocity under varying friction anisotropy conditions. Our analysis assumes a uniform weight distribution and negligible body rigidity, though the framework can be extended to more complex scenarios. Furthermore, we demonstrate the applicability of this approach to the undulatory motion of other elongated bodies in various media, where interactive forces can be described using resistive force theory, such as swimming through sand or viscous fluids.

Figures

Figures reproduced from arXiv: 2412.03725 by the authors.

Figure 1
Figure 1. Kinematic parameters of snake locomotion: (a) schematics of snake locomotion, (b) Velocity vectors [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. (a) Maximum efficiency η and (b) maximum velocity U for different friction ratio (µ), and the corresponding wave length (λ) and body angle (θ) for each case. The angles are in radian 3.2 Optimal Speed (U) The same method can be used to find the maximum velocity. The total velocity is the average of the local velocity of the body points over one cycle. The magnitude of U is expressed in integral form in equation 9. A… view at source ↗
Figure 3
Figure 3. Comparison of optimization results and wave shapes from numerical and analytical methods: (a) cost of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

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Reference graph

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