Optimization Study on Swimmers' Velocity Distribution Based on Euler-Lagrange Method

Abstract

In addressing the limitations of conventional experience-based speed allocation in competitive swimming, the objective of this study is to develop optimal strategies from a biomechanical and metabolic perspective, leveraging advanced tools such as mathematical modeling. Specifically, the forces on swimmers during freestyle at different stages were analyzed, and energy consumption was assessed with the ATP-CP model. The analysis included energy expenditure, speed limits, and collision losses. Subsequently, an optimization model for speed allocation was developed using the Euler-Lagrange method and Particle Swarm Optimization (PSO). The results show that in the 50-meter race, swimmers achieve a 0.3-second improvement when they rapidly attain and maintain their maximum sprint speed. For the 100-meter event, the initiation of a rapid increase in speed during the first half, followed by the maintenance of a steady pace, results in a 0.7-second enhancement. In the 200-meter race, a more uniform distribution of speed, coupled with a slight acceleration towards the finish line, results in a performance enhancement of 1.2 seconds. Furthermore, numerical simulations demonstrate that optimal strategies can reduce average power consumption by approximately 15% in comparison to conventional methods. These findings provide scientific support for the effectiveness of optimized speed distribution in improving athlete performance.