Variational problems in the works of academician O. I. Somov. Brachistochrone and tautochrone

The article presents an analysis of solutions to variational problems of mechanics in the works of Academician O.I. Somov (1815-1876). In 1869 O.I. Somov not only simplifies the solution to Abel's problem, but also gives a fundamental conclusion about extending the tautochrone problem from the gravity field to any potential field. The article shows how Somov, without using Euler integrals, finds the arc traversed by a body as a function of height, in the case when time does not depend on height (tautochrone). The author of the article examines in detail how, in a kinematic and dynamic problem, Somov immediately abandons Cartesian coordinates, switching to polar coordinates, saving the reader from endless substitutions.