Liouvillian solutions in the problem of rolling of a heavy homogeneous ball on a surface of revolution
DOI:
https://doi.org/10.21638/spbu01.2021.411Abstract
The problem of rolling without sliding of a homogeneous ball on a fixed surface under the action of gravity is a classical problem of nonholonomic system dynamics. Usually, when considering this problem, following the E. J. Routh approach it is convenient to define explicitly the equation of the surface, on which the ball’s centre is moving. This surface is equidistant to the surface, over which the contact point is moving. From the classical works of E. J. Routh and F. Noether it was known that if the ball rolls on a surface such that its centre moves along a surface of revolution, then the problem is reduced to solving the second order linear differential equation. Therefore it is interesting to study for which surface of revolution the corresponding second order linear differential equation admits Liouvillian solutions. To solve this problem it is possible to apply the Kovacic algorithm to the corresponding second order linear differential equation. In this paper we present our own method to derive the corresponding second order linear differential equation. In the case when the centre of the ball moves along the ellipsoid of revolution we prove that the corresponding second order linear differential equation admits a liouvillian solution.Keywords:
rolling without sliding, homogeneous ball, surface of revolution, Kovacic algorithm, Liouvillian solutions
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References
Литература
1. Routh E.J. The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies: Being Part II of a Treatise on the Whole Subject. Cambridge, Cambridge Univ. Press (2013). https://doi.org/10.1017/CBO9781139237284
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References
1. Routh E.J. The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies: Being Part II of a Treatise on the Whole Subject. Cambridge, Cambridge Univ. Press (2013). https://doi.org/10.1017/CBO9781139237284
2. Noether F. Uber rollende Bewegung einer Kugel auf Rotationsfl¨ ¨ achen. Leipzig, Teubner (1909).
3. Rashevskii P.K. Course of differential geometry. Moscow, Leningrad, GITTL Publ. (1950). (In Russian)
4. Kovacic J. An algorithm for solving second order linear homogeneous differential equations. J. Symb. Comput. 2, iss. 1, 3–43 (1986).
5. Kuleshov A.S., Chernyakov G.A. Application of the Kovacic algorithm for investigation of the problem of motion of a heavy body of revolution on a perfectly rough plane. Vestnik of Saint Petersburg University. Series 1. Mathematics. Mechanics. Astronomy, iss. 4, 93–102 (2013). (In Russian)
6. Kuleshov A.S., Itskovich M.O. Nonexistence of Liouvillian solutions in the problem of motion of a rotationally symmetric ellipsoid on a perfectly rough plane. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 4 (62), iss. 3, 291–299 (2017). https://doi.org/0.21638/11701/spbu01.2017.213 (In Russian) [Engl. transl.: Vestnik St. Petersb. Univ. Math. 50, 173–179 (2017). https://doi.org/10.3103/S106345411702008X].
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Published
2022-01-04
How to Cite
Kuleshov, A. S., & Solomina, D. V. (2022). Liouvillian solutions in the problem of rolling of a heavy homogeneous ball on a surface of revolution. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(4), 653–660. https://doi.org/10.21638/spbu01.2021.411
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Mechanics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
