Variational problems in the works of academician O. I. Somov. Brachistochrone and tautochrone
https://doi.org/10.22405/2226-8383-2024-25-5-216-227
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.
About the Author
Anna Olegovna Yulina
Saint Petersburg State University of Architecture and Civil Engineering; Peter the Great Saint Petersburg Polytechnic University
Russian Federation
Russian Federation
References
1. Yulina, A.O. 2023, “O.I. Somov’s tautochrona ”, History and pedagogy of natural science, № 2, pp. 41–44.
2. Somov, O.I. 1872, “Rational mechanics”, Kinematics. St. Petersburg. Printing house of the Imperial Academy of Sciences., 491 p.
3. Yulina, A.O. 2023, “O.I. Somov’s Mechanics”, History of science and technology, № 2, pp. 3–7.
4. Yulina, A.O. 2023, “Vector calculus in Somov’s mechanics”, History of science and technology, № 3, pp. 26–33.
5. Abel, N.H. 1823, “Solution de quelques probl`emes `a l’aide d’int´egrales d´efinies”, Opl¨osning af et Par Opgaver ved. Hjelp af bestemte Integraler, 1823. p. 11–27.
6. Abel, N.H. 1826, “R`esolution D’un Probl`eme de mecanique”, Journal f¨ur die reine und angewandte Mathematik, pp. 97–101.
7. Somov, O.I. 1866, “On the solution of a problem in mechanics proposed by Abel”, Notes of
8. the Imperial Academy of Sciences. St. Petersburg: Printing house of the Imperial Academy of
9. Sciences, Vol. 9, book 1, Separate pagination.
Review
For citations:
Yulina A.O. Variational problems in the works of academician O. I. Somov. Brachistochrone and tautochrone. Chebyshevskii Sbornik. 2024;25(5):216-227. (In Russ.) https://doi.org/10.22405/2226-8383-2024-25-5-216-227
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