Abstract
The motion of a heavy rigid thin rod on the surface of a right circular cylinder is considered. It is assumed that the angle between the generatrix of the cylinder and the direction of gravity is nonzero. The positions of equilibria of the rod on a cylinder form an equilibrium manifold (for all these equilibria the rod rests on the cylinder by its center of mass). The effect of transgression (nontrivial evolution along the equilibrium manifold) of the rod on the cylinder is studied using the normal form method.
Similar content being viewed by others
REFERENCES
Ya. V. Tatarinov, “Composition of nonlinear oscillations with evolution in the vicinity of equilibrium manifolds of reversible systems,” Vestn. Mosk. Univ., Ser. 1: Mat., Mekh. 5, 93–95 (1990).
Ya. V. Tatarinov, “Consequences of nonintegrable perturbations of integrable constraints: Nonlinear effects of motion near the equilibrium manifold,” J. Appl. Math. Mech. 56, 507–517 (1992).
A. D. Bruno, Local Methods in Nonlinear Differential Equations (Nauka, Moscow, 1979; Springer-Verlag, Berlin, 1989).
A. D. Bryuno, “Analytical form of differential equations I,” Tr. Mosk. Mat. O-va. 25, 119–262 (1971).
A. D. Bryuno, “Analytical form of differential equations. II,” Tr. Mosk. Mat. O-va. 26, 199–239 (1972).
V. F. Edneral, “Looking for periodic solutions of ODE Systems by the normal form method,” in Differential Equations with Symbolic Computation. Trends in Mathematics, Ed. by D. Wang, and Z. Zheng (Birkhäuser, Basel, 2005), pp. 173–200.
Ya. V. Tatarinov, “Consequences of nonintegrable perturbations of integrable constraints: Model problems of low dimensionality,” J. Appl. Math. Mech. 51, 579–586 (1987).
A. S. Kuleshov and I. I. Ulyatovskaya, “The transgression effect in the problem of motion of an al- most holonomic pendulum,” Vestn. S.-Peterb. Univ., Ser. 1: Mat., Mekh., Astron. 7(65), 356–360 (2020). https://doi.org/10.21638/11701/spbu01.2020.217
A. S. Kuleshov and S. V. Ifraimov, “Motion of the rod on a convex surface,” Vestn. S.-Peterb. Univ., Ser. 1: Mat., Mekh., Astron., No. 2, 105–110 (2013).
A. V. Karapetyan and V. V. Rumyantsev, “Stability of conservative and dissipative system,” in Proceedings of Science and Technology. General Mechanics (VINITI, Moscow, 1983), Vol. 6 [in Russian]; A. V. Karapetyan and V. V. Rumyantsev, “Stability of conservative and dissipative system,” in Applied Mechanics. Soviet Reviews. Stability and Analytical Mechanics (Hemisphere, New York, 1990), Vol. 1, pp. 1–145.
P. V. Voronets, “On equations of motion of nonholonomic systems,” Mat. Sb. 22, 659–686 (1901).
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors declare that they have no conflicts of interest.
Additional information
Translated by M. Shmatikov
About this article
Cite this article
Kuleshov, A.S., Vidov, N.M. Transgression Effect in the Problem of the Motion of a Rod on a Cylinder. Vestnik St.Petersb. Univ.Math. 56, 403–411 (2023). https://doi.org/10.1134/S1063454123030068
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1063454123030068