Geodesic
Variational problems in the works of academician O. I. Somov. Brachistochrone and tautochrone
Čebyševskij sbornik, Tome 25 (2024) no. 5, pp. 216-227 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Mots-clés : Variation, tautochrone
Keywords: variational problem, Abel problem. 
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     title = {Variational problems in the works of academician {O.~I.~Somov.} {Brachistochrone} and tautochrone},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {216--227},
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}
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A. O. Yulina. Variational problems in the works of academician O. I. Somov. Brachistochrone and tautochrone. Čebyševskij sbornik, Tome 25 (2024) no. 5, pp. 216-227. http://geodesic.mathdoc.fr/item/CHEB_2024_25_5_a13/