Optimization models for the force and energy in competitive sports

Abstract

A general optimal control model for running and swimming is developed, whose principal parameters are the energy reserves, the breathing rate and the force. This model is then applied to the world records in these disciplines. It describes the data very well and the parameters employed have values, which agree closely with those obtained by other means.

A mathematical model for the force and energetics in competitive running

“…Thus we will have to solve the differential equations governing the behavior of this model and compare the solutions with observations. In doing this we shall use as reference parameters Ve ~ 900-1200 J, Vr ~ 1400-1700 J, Me ~ 90-100W; Mo ~ 24-29W as well as (12), because these values are representative for good athletes [3].…”

“…Thus we shall neglect psychological factors and individual variations in build, biomechanics etc. This model is an extension of [3]. It is made up of three submodels, which describe the biomechanics, the energetics and the optimization.…”

“…III, see also [12,3], sprints are run mostly with maximal force, i.e. f = F. With this, the equations of motion (7) for t ~< to and (1) for t ~> to can readily be integrated in closed form.…”

“…With F = 6 . 9 5 (men), F = 6.30 (women) and b = q + 12c =0.75 (11) we obtain Contrary to [3] we have chosen a common t" 1 for men and women. The determination of F and r~ from the sprint data is complicated by the fact that both quantities have an opposite effect on the results.…”

Diagnosis in track and field performances past the 1988 Ben Johnson doping scandal

“…Since we nowadays assume that there is such a limit performance, here some scienti"c predecessors: Torre [19] tried to calculate such limits out of maximum force and energy consumption of a human being, with utopian results like 11 : 18 min for the 5000 m distance. A more sophistiated model in this direction is used by Behncke [2] calculating a possible world record for the same distance of 13 : 00.42 min, more realistic, but already beaten. Jokl [10] points out psychological arguments for the supposed limit in men's long jump of 8.90 m, already reached in 1968.…”