On a strict mathematical definition of the brachistochrone’s shape taking into account the thermal effect in the contact zone

Background. Current research on the properties of brachistochrones as lines with extreme properties indicates that this work is relevant because it takes into account for the first time the influence of thermal effects on the shape of the brachistochrone. The purpose of the study is an analytically rigorous solution to the problem. Matherials and methods. The main method for solving the problem is the moving basis method, which has proven itself well in solving many problems related to the study of various properties of brachistochrones. Results. A rigorous analytical solution of the formulated problem is given, taking into account the thermal effect, which was taken into account by introducing a dissipative function. Conclusions. Thanks to the algorithm proposed in the article, a general methodological approach has been formulated that is useful in solving such problems related to taking into account the thermal properties of materials.