Modelling the optimization of world-class 400†m and 1,500†m running performances using high-resolution data.

The 400 m and 1,500 m are track events that rely on different but important contributions from both the aerobic and anaerobic energy systems. The purpose of this study is to model men's and women's 400 m and 1,500 m championship performances to gain a deeper understanding of the key mechanical and physiological factors affecting running speed and bend running using high-resolution data from live competition (10 Hz). To investigate World-class athletes' instantaneous speeds, propulsive forces and aerobic and anaerobic energy, we model and simulate the performances of the men's and women's European Athletics 400 m champions, Matthew Hudson-Smith and Femke Bol, as well as the men's European Athletics 1,500 m champion, Jakob Ingebrigtsen, and the women's European Athletics U23 1,500 m champion, Gaia Sabbatini. The simulations show that a fast start is essential in both the 400 m and 1,500 m because of the need for fast oxygen kinetics, with peak running speeds occurring within the first ∼50 m in both events. Subsequently, 400 m athletes slow continually from this maximum speed to the finish, and a total anaerobic contribution of ∼77% is found for both male and female champions. The key to faster 400 m racing is to reduce the decrease in velocity: this comes from both a high VO2 and a high anaerobic contribution. Ingebrigtsen's winning tactic in the European 1,500 m final is to adopt a very fast cruising pace from 300 m onwards that is possible because he is able to maintain a high VO2 value until the end of the race and has a large anaerobic contribution. He has fast VO2 kinetics that does not require as fast a start as his opponents, but then he speeds up in the last two laps, without a fast sprint finish. The comparison between Sabbatini's slower and quicker races (∼8 s difference) shows that it is the improvement of aerobic metabolism that has the greatest effect on 1,500 m performance. Coaches should note in particular that the all-out pacing nature of the 400 m requires the prioritization of anaerobic energy system development, and those who coach the 1,500 m should note the differing energy contributions between even-paced races and championship racing.

Pace and motor control optimization for a runner.

We present a model which encompasses pace optimization and motor control effort for a runner on a fixed distance. We see that for long races, the long term behaviour is well approximated by a turnpike problem, that allows to define an approximate optimal velocity. We provide numerical simulations quite consistent with this approximation which leads to a simplified problem. The advantage of this simplified formulation for the velocity is that if we have velocity data of a runner on a race, and have access to his [Formula: see text], then we can infer the values of all the physiological parameters. We are also able to estimate the effect of slopes and ramps.

Optimal speed in Thoroughbred horse racing.

The objective of this work is to provide a mathematical analysis on how a Thoroughbred horse should regulate its speed over the course of a race to optimize performance. Because Thoroughbred horses are not capable of running the whole race at top speed, determining what pace to set and when to unleash the burst of speed is essential. Our model relies on mechanics, energetics (both aerobic and anaerobic) and motor control. It is a system of coupled ordinary differential equations on the velocity, the propulsive force and the anaerobic energy, that leads to an optimal control problem that we solve. In order to identify the parameters meaningful for Thoroughbred horses, we use velocity data on races in Chantilly (France) provided by France Galop, the French governing body of flat horse racing in France. Our numerical simulations of performance optimization then provide the optimal speed along the race, the oxygen uptake evolution in a race, as well as the energy or the propulsive force. It also predicts how the horse has to change its effort and velocity according to the topography (altitude and bending) of the track.

Correction to 'How to build a new athletic track to break records'.

[This corrects the article DOI: 10.1098/rsos.200007.].

How to build a new athletic track to break records.

We introduce a new optimal control model which encompasses pace optimization and motor control effort for a runner on a fixed distance. The system couples mechanics, energetics, neural drive to an economic decision theory of cost and benefit. We find how effort is minimized to produce the best running strategy, in particular, in the bend. This allows us to discriminate between different types of tracks and estimate the discrepancy between lanes. Relating this model to the optimal path problem called the Dubins path, we are able to determine the geometry of the optimal track and estimate record times.

A Model for World-Class 10,000 m Running Performances: Strategy and Optimization.

The distribution of energetic resources in world-class distance running is a key aspect of performance, with athletes relying on aerobic and anaerobic metabolism to greater extents during different parts of the race. The purpose of this study is to model 10,000 m championship performances to enable a deeper understanding of the factors affecting running speed and, given that more than half the race is run on curves, to establish the effect of the bends on performance. Because a limitation of time split data is that they are typically averaged over 100-m or 1,000-m segments, we simulate two 10,000 m runners' performances and thus get access to their instantaneous speed, propulsive force and anaerobic energy. The numerical simulations provide information on the factors that affect performance, and we precisely see the effect of parameters that influence race strategy, fatigue, and the ability to speed up and deal with bends. In particular, a lower anaerobic capacity leads to an inability to accelerate at the end of the race, and which can accrue because of a reliance on anaerobic energy to maintain pace in an athlete of inferior running economy. We also see that a runner with a worse running economy is less able to speed up on the straights and that, in general, the bends are run slower than the straights, most likely because bend running at the same pace would increase energy expenditure. Notwithstanding a recommendation for adopting the accepted practices of improving aerobic and anaerobic metabolism through appropriate training methods, coaches are advised to note that athletes who avoid mid-race surges can improve their endspurt, which are the differentiating element in closely contested championship races.

Optimizing running a race on a curved track.

In order to determine the optimal strategy to run a race on a curved track according to the lane number, we introduce a model based on differential equations for the velocity, the propulsive force and the anaerobic energy which takes into account the centrifugal force. This allows us to analyze numerically the different strategies according to the types of track since different designs of tracks lead to straights of different lengths. In particular, we find that the tracks with shorter straights lead to better performances, while the double bend track with the longest straight leads to the worst performances and the biggest difference between lanes. Then for a race with two runners, we introduce a psychological interaction: there is an attraction to follow someone just ahead, but after being overtaken, there is a delay before any benefit from this interaction occurs. We provide numerical simulations in different cases. Overall, the results agree with the IAAF rules for lane draws in competition, where the highest ranked athletes get the center lanes, the next ones the outside lanes, while the lowest ranked athletes get the inside lanes.

Nonclassical rotational inertia of a supersolid.

As proposed by Leggett [Phys. Rev. Lett. 25, 1543 (1970)10.1103/PhysRevLett.25.1543], the supersolidity of a crystal is characterized by the nonclassicalical Rotational Inertia (NCRI) property. Using a model of quantum crystal introduced by Josserand, Pomeau, and Rica [Phys. Rev. Lett. 72, 2426 (1994)10.1103/PhysRevLett.72.2426], we prove that NCRI occurs. This is done by analyzing the ground state of the aforementioned model, which is related to a sphere packing problem, and then deriving a theoretical formula for the moment of inertia. We infer a lower estimate for the NCRI fraction, which is a landmark of supersolidity.

Dissipative flow and vortex shedding in the Painlevé boundary layer of a Bose-Einstein condensate.

This paper addresses the drag force and formation of vortices in the boundary layer of a Bose-Einstein condensate stirred by a laser beam following the experiments of Phys. Rev. Lett. 83, 2502 (1999)]. We make our analysis in the frame moving at constant speed where the beam is fixed. We find that there is always a drag around the laser beam. We also analyze the mechanism of vortex nucleation. At low velocity, there are no vortices and the drag has its origin in a wakelike phenomenon: This is a particularity of trapped systems since the density gets small in an extended region. The shedding of vortices starts only at a threshold velocity and is responsible for a large increase in drag. This critical velocity for vortex nucleation is lower than the critical velocity computed for the corresponding 2D problem at the center of the cloud.