On detachment conditions in the problem on the motion of a rigid body on a rough plane
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In the absence of friction, the detachment conditions on stationary motions do not hold. However, if the angle θ between the symmetry axis and the vertical decreases to zero, motions close to stationary motions are necessarily accompanied by detachments.
The same conclusion holds for a thin disk that rolls on the support without sliding.
For a disk of nonzero thickness in the absence of sliding, the detachment conditions hold on stationary motions in some domain in the space of parameters; in this case, the angle θ is not less than 49 degrees. For small values of θ, the contact between the body and the support does not break in a neighborhood of stationary motions.
Key wordsunilateral constraint friction Painlevé paradoxes
MSC2000 numbers70E18 70E50 70G70
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- 1.Appell, P., Traité de Mécanique Rationelle, Paris: Gauthier-Villars, 1953.Google Scholar
- 2.Markeev, A.P., Dynamics of a Body Being Contiguous to a Rigid Surface, Moscow: Nauka, 1991 (in Russian)Google Scholar
- 3.Routh, E.J., Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies, London: MacMillan, 1905.Google Scholar
- 5.Karapetyan, A.V. and Rumjantsev, V.V., Stability of Conservative and Dissipative Systems, in Appl. Mech. Soviet Reviews. V. 1. Stability and Analytical Mechanics, Mikhailov, G.K. and Parton V.Z., Eds., New York: Hemisphere, 1990, pp. 1–144.Google Scholar
- 8.Painlevé, P., Leçon sur le Frottement, Paris: Hermann, 1885.Google Scholar
- 11.Kozlov, V.V. and Treshchev, D.V., Billiards, Providence: Amer. Math. Soc., 1991.Google Scholar
- 15.Deryabin, M.V., General principles of dynamics and the theory of unilateral constraints, Vestnik Mockov. Univ. Ser. I Mat. Mekh., 1998, no. 1, pp. 53–59.Google Scholar
- 16.Ivanov, A.P., On Shock-Free Jumps of a Non-Homogeneous Wheel, Izv. Ross. Akad. Nauk Ser. Mekh. Tverd. Tela, 1993, no. 1, pp. 61–64.Google Scholar
- 18.Caps, H., Dorbolo, S., Ponte, S., Croisier, H., and Vanderwalle, N., Rolling and Slipping Motion of Euler’s Disk, Phys. Rev. E, 2004, vol. 69, 056610 (6 pages).Google Scholar
- 19.Easwar, K., Rouyer, F., and Menon, N., Speeding to a Stop: the Finite-Time Singularity of a Spinning Disk, Phys. Rev. E, 2002, vol. 66, 045102 (3 pages).Google Scholar
- 20.McDonald, A.J. and McDonald, K.T., The Rolling Motion of a Disk on a Horizontal Plane, Preprint of Princeton High School, New Jersey, 2001.Google Scholar