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arxiv: 2410.02767 · v2 · pith:3UD2JJGFnew · submitted 2024-09-17 · ⚛️ physics.class-ph · cs.NA· math.NA

A mathematical model for Nordic skiing

Jane Shaw MacDonald , Rafael Ordo\~nez Cardales , John M. Stockie This is my paper

Pith reviewed 2026-05-23 20:41 UTC · model grok-4.3

classification ⚛️ physics.class-ph cs.NAmath.NA
keywords Nordic skiingmathematical modelordinary differential equations3D space curvenumerical simulationNewton's lawssports science
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The pith

A model using 3D curves and Newton's laws predicts skier motion on real Nordic courses.

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

The paper presents a mathematical model for Nordic skiing in which the course is a three-dimensional space curve and the skier's dynamics are governed by a nonlinear system of ordinary differential equations derived from Newton's laws of motion. Forces including gravity, friction, and air resistance are incorporated into the equations. An algorithm using Hermite spline interpolation for the curve, numerical quadrature, and a high-order ODE solver generates simulations of the skier's path and speed. These simulations are validated by comparison with measurements taken from skiers on actual courses. This work shows how undergraduate-level calculus and computing can model a complex sport and yield practical insights for sports science.

Core claim

The motion of a Nordic skier along a three-dimensional course can be accurately simulated by solving a nonlinear ODE system based on Newton's second law, where the course geometry is represented by a space curve, and the resulting numerical predictions match experimental data collected from real ski runs.

What carries the argument

Nonlinear system of ordinary differential equations derived from Newton's laws, with course geometry given by a three-dimensional space curve.

If this is right

  • The model can simulate how terrain variations affect a skier's speed and total time on course.
  • Changes in parameters like friction coefficients can be used to study the effects of different snow conditions or equipment.
  • The numerical method provides a way to analyze the contribution of individual forces to overall performance.
  • It serves as an example for applying mathematical techniques to study athletic activities.

Where Pith is reading between the lines

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

  • The approach could be adapted to model other endurance sports on variable terrain such as trail running or cycling.
  • Simulations might help coaches test the impact of different racing lines without physical trials.
  • Extending the model to include skier technique variations like poling or turning could provide more detailed performance analysis.

Load-bearing premise

The parameters chosen for the forces in the ODE system adequately represent the physical interactions between the skier, skis, snow, and air, and the 3D curve precisely captures the actual course geometry.

What would settle it

Collecting speed and position data from skiers on a previously unmodeled course and finding that the simulated times or velocities differ substantially from observations despite adjustments to model parameters.

Figures

Figures reproduced from arXiv: 2410.02767 by Jane Shaw MacDonald, John M. Stockie, Rafael Ordo\~nez Cardales.

Figure 1
Figure 1. Figure 1: Three views of the 4.2 km Ole course in Toblach [8]. (a, Top Left) 3D plot of the skier path defined [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of three splines using both elevation (a,left) and plane views (b,right): linear (dotted [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The final 250 m stretch of the Ole course, emphasizing the spurious oscillations that can be [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) A 2D elevation profile with the course parameterized as [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Simulations of the MSH baseline case. (a, Top Left) Elevation profile shown in terms of ( [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Simulations of the Welde course, taking all parameters equal to the baseline values except [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Simulation of the 4.2 km Ole course with the baseline parameters, displaying speed and projected [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Definition of the inclination angle θ (measured relative to the horizontal plane) and azimuth angle φ (in the x, y-plane, measured relative to the x-axis) for a 3D ski course parameterized as ⃗r(ξ) = [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The difference in projected arc length ξ between the 2D and 3D simulations on the Ole course with three choices of braking parameter (γ = 0, γstep, γskid ). The braking threshold is set to ac,min = 2 and all other parameters are taken from the MSH baseline test. and generate a moderate braking force with Fb ≈ 1. In contrast, the similar curvature near the base of the hill occurs once the course has flatten… view at source ↗
Figure 10
Figure 10. Figure 10: (a, Top) The braking force is displayed along the scaled Ole course profile in a 3D view of the [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
read the original abstract

Nordic skiing provides fascinating opportunities for mathematical modelling studies that exploit methods and insights from physics, applied mathematics, data analysis, scientific computing and sports science. A typical ski course winds over varied terrain with frequent changes in elevation and direction, and so its geometry is naturally described by a three-dimensional space curve. The skier travels along a course under the influence of various forces, and their dynamics can be described using a nonlinear system of ordinary differential equations (ODEs) that are derived from Newton's laws of motion. We develop an algorithm for solving the governing equations that combines Hermite spline interpolation, numerical quadrature and a high-order ODE solver. Numerical simulations are compared with measurements of skiers on actual courses to demonstrate the effectiveness of the model. Throughout, we aim to illustrate how elementary concepts from undergraduate courses in calculus and scientific computing can be applied to study real problems in sport, which we hope will provide stimulating examples for both instructors and students. At the same time, we demonstrate how these concepts are capable of providing novel insights into skiing that should also be of interest to sport scientists.

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.

Referee Report

2 major / 2 minor

Summary. The paper models Nordic skiing by representing the course as a 3D space curve and deriving a nonlinear ODE system from Newton's laws that incorporates gravity, friction, air resistance, and other forces. It develops a numerical algorithm using Hermite spline interpolation for the curve, numerical quadrature, and a high-order ODE solver. Simulations are compared to measurements of skiers on real courses to demonstrate model effectiveness, while illustrating applications of undergraduate calculus and scientific computing to sports science.

Significance. If the comparisons hold with independently determined parameters and quantitative error metrics, the model could serve as a reproducible framework for analyzing skier dynamics on complex terrain and as an educational bridge between elementary numerical methods and real sports data. The explicit use of Newton's laws plus measured course geometry, rather than purely empirical fitting, is a potential strength for predictive use in course design or technique analysis.

major comments (2)
  1. [Results/comparison section] Results/comparison section: the abstract and introduction state that simulations are compared with measurements on actual courses to demonstrate effectiveness, yet no quantitative fit metrics (RMS error, R², mean absolute deviation in speed or position) or uncertainty quantification are provided. This directly affects the central claim that the model is shown to be effective.
  2. [Model formulation and parameter section] Model formulation and parameter section: the ODE system includes several physical coefficients (friction, drag, possibly power output). It is not stated whether these are obtained from separate independent experiments or adjusted to the trajectory data used for validation. If the latter, the reported agreement tests calibration rather than a priori prediction from Newton's laws plus the 3D geometry alone.
minor comments (2)
  1. [Numerical algorithm] The description of the Hermite spline interpolation and quadrature steps could include a brief pseudocode or explicit reference to the specific quadrature rule employed for arc-length computation.
  2. [Figures] Figure captions for the course curve and trajectory plots should explicitly state the source of the measured data (e.g., GPS sampling rate, skier mass, snow conditions) to allow reproducibility assessment.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below and will revise the paper to improve clarity and strengthen the validation claims.

read point-by-point responses

  1. Referee: [Results/comparison section] Results/comparison section: the abstract and introduction state that simulations are compared with measurements on actual courses to demonstrate effectiveness, yet no quantitative fit metrics (RMS error, R², mean absolute deviation in speed or position) or uncertainty quantification are provided. This directly affects the central claim that the model is shown to be effective.

    Authors: We agree that the absence of quantitative error metrics limits the strength of the validation claim. The current manuscript presents comparisons via figures of simulated versus measured speeds and positions but does not report RMS errors, R², or similar statistics. In the revised version we will compute and tabulate these metrics (RMS error and mean absolute deviation in speed and along-track position) for the real-course data sets, along with a short discussion of measurement uncertainty. revision: yes

  2. Referee: [Model formulation and parameter section] Model formulation and parameter section: the ODE system includes several physical coefficients (friction, drag, possibly power output). It is not stated whether these are obtained from separate independent experiments or adjusted to the trajectory data used for validation. If the latter, the reported agreement tests calibration rather than a priori prediction from Newton's laws plus the 3D geometry alone.

    Authors: The manuscript does not currently state the origin of the physical coefficients. We will add an explicit subsection (or table) listing each coefficient together with its source and reference to independent experimental or literature values. If any coefficient was informed by the validation trajectories, we will note this and qualify the interpretation of the comparisons as a combination of prediction and calibration. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation from Newton's laws with external validation

full rationale

The paper derives a nonlinear ODE system directly from Newton's laws for forces on the skier along a 3D space curve, develops a numerical solver using Hermite splines and quadrature, and validates by comparing simulations to independent measurements of skiers on real courses. No quoted steps reduce predictions to fitted inputs by construction, no self-citation load-bearing for central claims, and no ansatz or uniqueness imported from prior author work. The central claim remains independent of the validation data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides insufficient detail to enumerate specific free parameters or invented entities; the model relies on standard Newtonian mechanics.

axioms (1)
  • domain assumption Newton's laws of motion apply to describe skier dynamics under the listed forces
    Explicitly stated as the basis for deriving the nonlinear ODE system.

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