Application of Lagrange multiplier for time-trial strategies

If one wishes to minimize the time of a time trial whose one portion is with—and another portion against—the wind, the question arises on how to optimize one’s effort; for instance, should the power or speed be kept constant? Herein, to address such questions, we invoke the Lagrange multipliers.
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Danek, T., Slawinski, M.A., Stanoev, T. (2020) On modelling bicycle power-meter measurements: Part II. Relations between rates of change of model quantities, 2005.04480 [physics.pop-ph]