Equations of motion of a track cyclist: Active particle dynamics under strong geometric constraints

Using in situ measurements from the 2022 World Championships, we derive the equations of motion for a track cyclist as a coupled bicycle-rider-track system. A key observation is that instantaneous pedal power does not map directly onto bicycle speed because elastic compliance within the bicycle-rider assembly stores and releases energy over the crank cycle, introducing a phase lag between input power and translational motion. To resolve this, we develop a phase-averaging framework over the pedaling period that yields an effective, stiff dynamical system driven by cycle-averaged power. The resulting reduced model predicts the evolution of speed without requiring high-frequency pedal-level inputs. A central feature is the geometric coupling between rider dynamics and velodrome shape: the equations link wheel-frame velocity to the center-of-mass velocity through the local track curvature, accounting for banking and radius variations along the lap. Applied to competition data, the framework reproduces the measured speed evolution across straights and turns. Beyond track cycling, the phase-average method provides a general route to modeling propulsion in compliant, cyclic systems where input power is periodic but motion is governed by averaged dynamics.