# Brachistochrone on a Velodrome

## TL;DR

This paper extends the classical Brachistochrone problem to find optimal cycling trajectories on velodrome surfaces, combining analytical and numerical methods, and considers effects like fatigue to improve lap time predictions.

## Contribution

It introduces a novel approach to determine the fastest cycling path on velodromes by modeling the cyclist as an active particle and solving the problem on real track geometries.

## Key findings

- Analytical solutions for simple geometries like slopes and cones.
- Numerical solutions on actual velodrome surface.
- Discussion of fatigue effects on optimal trajectories.

## 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.