A mathematical model for Nordic skiing

TL;DR

This paper presents a mathematical model of Nordic skiing that uses physics, applied mathematics, and scientific computing to simulate skier dynamics on varied terrain, validated with real measurements.

Contribution

It develops a novel algorithm combining spline interpolation and high-order ODE solving to accurately simulate skier motion on complex courses.

Findings

Numerical simulations align well with actual skier data.

Elementary calculus concepts can effectively model complex sports dynamics.

The model offers new insights into skiing physics and performance.

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.