Stability of steady rotations in the nonholonomic Routh problem
- 34 Downloads
We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball at the upmost, downmost and saddle point.
Key wordsnonholonomic constraint stationary rotations stability
MSC2000 numbers34D20 70E40 37J35
Unable to display preview. Download preview PDF.
- 1.Routh, E.J., Advanced Dynamics of a System of Rigid Bodies, 6th ed., London: MacMillan Company, 1905. Reprinted by New York: Dover Publications, 1955.Google Scholar
- 6.Yaroshchuk, V.A., New Cases of the Existence of an Integral Invariant in a Problem on the Rolling of a Rigid Body, Without Slippage, on a Fixed Surface, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1992, no. 6, pp. 26–30 (in Russian).Google Scholar
- 7.Routh, E.J., A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion (Adams Price Essay), London: MacMillan Company, 1877.Google Scholar
- 10.Glukhikh, Yu.D., Tkhai, V.N., Chevallier, D.P., On the Stability of Permanent Rotations of a Heavy Homogeneous Ellips oid on an Ideally Rough Plane, in Problems in the Investigation of the Stability and Stabilization of Motion, Part I, Ross. Akad. Nauk, Vychisl. Tsentr im. A. A. Dorodnitsyna, Moscow, 2000, pp. 87–104 (in Russian).Google Scholar
- 11.Moser, J.K., Lectutes on Hamiltonian Systems, Memoirs Am. Math. Soc., 1968, vol. 81, pp. 1–60.Google Scholar
- 12.Arnold, V.I., Geometrical Methods in the Theory of Ordinary Differential Equations, New York: Springer-Verlag, 1988.Google Scholar
- 13.Chaplygin, S.A., On a Paraboloid Pendulum, 1898; reprinted in: Polnoe sobranie sochinenii (Collected Works), Leningrad: Akad. Nauk SSSR, 1933, vol. 1, pp. 194–199.Google Scholar