Integrability by quadratures of the problem of rolling motion of a heavy homogeneous ball on a surface of revolution of the second order

Authors

  • Alexander S. Kuleshov Lomonosov Moscow State University, 1, Leninskie Gory, Moscow, 119991, Russian Federation
  • Alexander A. Shishkov Lomonosov Moscow State University, 1, Leninskie Gory, Moscow, 119991, Russian Federation

DOI:

https://doi.org/10.21638/spbu01.2024.208

Abstract

In this paper, we consider the problem of rolling of a heavy homogeneous ball on a perfectly rough surface of revolution. Usually, when considering this problem, it is convenient to specify explicitly the surface along which the center of the ball moves during rolling, instead of the surface along which the ball rolls. The surface on which the center of the ball moves is equidistant to the surface on which the ball is rolling. It is well known, that the considered problem is reduced to the integration the second order linear homogeneous differential equation. In this paper we assume, that the surface along which the center of the ball moves is a non - degenerate surface of revolution of the second order. Using the Kovacic algorithm we prove that the general solution of the corresponding linear differential equation can be found explicitly. This means, that in this case the problem of rolling of a ball on a surface of revolution can be integrated by quadratures.

Keywords:

rolling without sliding, homogeneous ball, surface of revolution of the 2nd order, integrability by quadratures

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References

Литература

1. Кулешов А. С., Соломина Д. В. Лиувиллевы решения в задаче о качении тяжелого однородного шара по поверхности вращения. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 8 (66), вып. 4, 653-660 (2021). https://doi.org/10.21638/spbu01.2021.411

2. 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 University Press (2013). https://doi.org/10.1017/CBO9781139237284

3. Noether F. Uber rollende Bewegung einer Kugel auf Rotationsfl¨ ¨ achen. Leipzig, Teubner (1909).

4. Kovacic J. An algorithm for solving second order linear homogeneous differential equations. J. Symb. Comput. 2 (1), 3-43 (1986). https://doi.org/10.1016/S0747-7171(86)80010-4

5. Кулешов А. С., Черняков Г. А. Применение алгоритма Ковачича для исследования задачи о движении тяжелого тела вращения по абсолютно шероховатойплоскости. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 4, 93-102 (2013).

6. Кулешов А. С., Ицкович М. О. Несуществование лиувиллевых решенийв задаче о движении эллипсоида вращения по абсолютно шероховатой плоскости. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 4 (62), вып. 2, 291-299 (2017). https://doi.org/0.21638/11701/spbu01.2017.213

References

1. Kuleshov A. S., Solomina D. V. 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 (66), iss. 4, 653-660 (2021). https://doi.org/10.21638/spbu01.2021.411 (In Russian) [Eng. transl.: Vestnik St. Petersburg University, Mathematics 54, iss. 4, 405-410 (2021). https://doi.org/10.1134/S1063454121040105].

2. 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 University Press (2013). https://doi.org/10.1017/CBO9781139237284

3. Noether F. Uber rollende Bewegung einer Kugel auf Rotationsfl¨ ¨ achen. Leipzig, Teubner (1909).

4. Kovacic J. An algorithm for solving second order linear homogeneous differential equations. J. Symb. Comput. 2 (1), 3-43 (1986). https://doi.org/10.1016/S0747-7171(86)80010-4

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. 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) [Eng. transl.: Vestnik St Petersburg University, Mathematics 50, iss. 2, 173-179 (2021). https://doi.org/10.3103/S106345411702008X].

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Published

2024-08-10

How to Cite

Kuleshov, A. S., & Shishkov, A. A. (2024). Integrability by quadratures of the problem of rolling motion of a heavy homogeneous ball on a surface of revolution of the second order. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 11(2), 347–353. https://doi.org/10.21638/spbu01.2024.208

Issue

Vol. 11 No. 2 (2024)

Section

Mechanics

License

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.

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