Two-factor optimization in the brachistochrone problem

The paper puts forward a new modification of the well-known brachistochrone problem. The joint account of minimizing the motion time and the trajectory length in their functional relationship has been introduced. A two-factor optimization criterion (TOC) was constructed in the form of a product of two particular criteria, which made it possible to find the best compromise between them. On the TOC basis a solution to the problem of a two-factor brachistochrone was obtained using a preliminary consideration of the auxiliary problem on a brachistochrone with a given length. A rational practical solution of the problem was proposed. It was characterized by a simpler geometry than the strictly optimal one: to adopt a circular arc with a central angle selected on the basis of the taken TOC.