Email this article 
About tautochronic movements
- Authors: Petrov A.G.1
-
Affiliations:
- Ishlinsky Institute of Mechanics Problems of RAS
- Issue: Vol 518, No 1 (2024)
- Pages: 22-28
- Section: MATHEMATICS
- URL: https://jdigitaldiagnostics.com/2686-9543/article/view/647985
- DOI: https://doi.org/10.31857/S2686954324040045
- EDN: https://elibrary.ru/YZNYQS
- ID: 647985
Cite item
Open Access
Access granted
Subscription Access
Abstract
It is shown that a material point, under the influence of an attractive linear force and a repulsive force inversely proportional to the cube of the distance from the center of attraction, performs a periodic motion, the period of which does not depend on the initial data (tautochronic motion). The problem is reduced to a nonlinear autonomous second-order equation, the general solution of which is expressed in terms of elementary functions. It has also been proven that for other power laws of repulsive force, except for degrees 0, 1 and –3, the movement of a material point is not tautochronous.
About the authors
A. G. Petrov
Ishlinsky Institute of Mechanics Problems of RAS
Author for correspondence.
Email: petrovipmech@gmail.com
Russian Federation, Moscow
References
- Аппель P. Теоретическая механика. Т. 1. Физ. мат. лит. М., 1960 / Appel P. Traité de mécanique rationnelle – Tome premier statique-dynamique du point 1902.
- Ландау Л.Д., Лифшиц Е.М. Механика. Физ. мат. лит. М., 1965
- Osypowski E.T., Olsson M.G. Isynchronous motion in classical mechanics // Am.J. Phys. 1987. V. 55. P. 720–725.
- Chalykh O.A., Veselov A.P. A Remark on Rational Isochronous Potentials Journal of Nonlinear Mathematical Physics Volume 12, Supplement 1. 2005. P. 179–183.
- Буданов В.М. Об одной изохронной нелинейной системе. // Вестн. моск. ун-та. сер.1, математика. механика. 2013. № 6. С. 59–63. / V.M. Budanov, On a nonlinear isochronous system. Moscow Univ. Mech. Bull. 68, (2013).
- Биркгоф Д.Д. Динамические системы. Ижевск; Издательский дом “Удмуртский университет”, 1999, 408 с. / Birkhoff D. Dynamical Systems. Publisher, Edwards, 1927
- Журавлев В.Ф. Основы теоретической механики. М.: ФИЗМАТЛИТ, 2008. 304 с./ Zhuravlev V.F. Fundamentals of Theoretical Mechanics. М.: FIZMATLIT, 2008. 304 с. (in Russian)
- Журавлев В.Ф. Инвариантная нормализация неавтономных гамильтоновых систем. // ПММ, 2002. Т. 66. Вып. 3. С. 356–365. / Zhuravlev V. Ph. Invariant normalization of non-autonomous Hamiltonian systems J. Applied Mathematics and Mechanics. 2004. V. 66. № 3. P. 356–365
Supplementary files
