The equations for the brachistochrone of a self-accelerating particle are found by solving a problem in the calculus of variations with a constraint and a free end condition then applied to find the shape of a running track.
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Communicated by D. Lawden
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Drummond, J.E., Downes, G.L. The brachistochrone with acceleration: A running track. J Optim Theory Appl 7, 444–449 (1971). https://doi.org/10.1007/BF00931980