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

A. O. Yulina<sup>ab

a</sup> Peter the Great Saint Petersburg Polytechnic University (St. Petersburg)
^b Saint Petersburg State University of Architecture and Civil Engineering (St. Petersburg)

Abstract: 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.

Keywords: Variation, tautochrone, variational problem, Abel problem.

UDC: 531.091

Received: 12.05.2024
Accepted: 26.12.2024

DOI: 10.22405/2226-8383-2024-25-5-216-227