An invariant measure and the probability of a fall in the problem of an inhomogeneous disk rolling on a plane. (English) Zbl 1410.37059
Summary: This paper addresses the problem of an inhomogeneous disk rolling on a horizontal plane. This problem is considered within the framework of a nonholonomic model in which there is no slipping and no spinning at the point of contact (the projection of the angular velocity of the disk onto the normal to the plane is zero). The configuration space of the system of interest contains singular submanifolds which correspond to the fall of the disk and in which the equations of motion have a singularity. Using the theory of normal hyperbolic manifolds, it is proved that the measure of trajectories leading to the fall of the disk is zero.
MSC:
| 37J60 | Nonholonomic dynamical systems |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 70F25 | Nonholonomic systems related to the dynamics of a system of particles |
Keywords:
nonholonomic mechanics; regularization; blowing-up; invariant measure; ergodic theorems; normal hyperbolic submanifold; Poincaré map; first integralsReferences:
| [1] | Abdullaev, S. S. and Zaslavskii, G. M., Classical Nonlinear Dynamics and Chaos of Rays in Problems of Wave Propagation in Inhomogeneous Media, Sov. Phys. Usp., 1991, vol. 34, no. 8, pp. 645-664; see also: Uspekhi Fiz. Nauk, 1991, vol. 161, no. 8, pp. 1-43. |
| [2] | Afonin, A.A. and Kozlov, V.V., Problem on Falling of Disk Moving on Horizontal Plane, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 1997, no. 1, pp. 7-13 (Russian). |
| [3] | Appell, P., Sur l’intégration des équations du mouvement d’un corps pesant de révolution roulant par une arete circulaire sur un plan horizontal: cas particulier du cerceau, Rend. Circ. Mat. Palermo, 1900, vol. 14, no. 1, pp. 1-6. · JFM 31.0693.02 |
| [4] | Arnol’d, V. I., Kozlov, V.V., and Neĭshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006. · Zbl 1105.70002 |
| [5] | Batista, M., Integrability of the Motion of a Rolling Disk of Finite Thickness on a Rough Plane, Internat. J. Non-Linear Mech., 2006, vol. 41, pp. 850-859. · Zbl 1160.70346 |
| [6] | Batista, M., The Nearly Horizontally Rolling of a Thick Disk on a Rough Plane, Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 344-354. · Zbl 1229.70014 |
| [7] | Bolsinov, A.V., Kilin, A.A., and Kazakov, A.O., Topological Monodromy As an Obstruction to Hamiltonization of Nonholonomic Systems: Pro or Contra?, J. Geom. Phys., 2015, vol. 87, pp. 61-75. · Zbl 1322.37029 |
| [8] | Borisov, A. V., Fedorov, Y.N., and Mamaev, I. S., Chaplygin ball over a fixed sphere: an explicit integration, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 557-571. · Zbl 1229.37080 |
| [9] | Borisov, A.V., Kazakov, A.O., and Kuznetsov, S.P., Nonlinear dynamics of the rattleback: a nonholonomic model, Sov. Phys. Usp., 2014, vol. 57, no. 5, pp. 453-460. |
| [10] | Borisov, A.V., Kazakov, A.O., and Sataev, I.R., The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top, Regul. Chaotic Dyn., 2014, vol. 19, no. 6, pp. 718-733. · Zbl 1358.70006 |
| [11] | Borisov, A. V.; Kilin, A. A.; Mamaev, I. S., Hamiltonicity and integrability of the Suslov problem (2011) · Zbl 1277.70008 |
| [12] | Borisov, A.V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443-490. · Zbl 1229.70038 |
| [13] | Borisov, A. V., and Mamaev I. S., Isomorphism and Hamilton representation of some nonholonomic systems, Siberian Math. J., 2007, vol. 48, no. 1, pp. 26-36. · Zbl 1164.37342 |
| [14] | Borisov, A.V. and Mamaev, I. S., The Nonexistence of an Invariant Measure for an Inhomogeneous Ellipsoid Rolling on a Plane, Math. Notes, 2005, vol. 77, no. 6, pp. 855-857; see also: Mat. Zametki, 2005, vol. 77, no. 6, pp. 930-932. · Zbl 1081.37539 |
| [15] | Borisov, A. V. and Mamaev, I. S., The Rolling Motion of a Rigid Body on a Plane and a Sphere: Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177-200. · Zbl 1058.70009 |
| [16] | Borisov, A.V., Mamaev, I. S., and Bizyaev, I.A., Dynamical Systems with Non-Integrable Constraints: Vaconomic Mechanics, Sub-Riemannian Geometry, and Non-Holonomic Mechanics, Russian Math. Surveys, 2017, vol. 72, no. 1, pp. 1-32; see also: Uspekhi Mat. Nauk, 2017, vol. 72, no. 5(437), pp. 3-62. |
| [17] | Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 3, pp. 277-328. · Zbl 1367.70007 |
| [18] | Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Jacobi Integral in Nonholonomic Mechanics, Regul. Chaotic Dyn., 2015, vol. 20, no. 3, pp. 383-400. · Zbl 1367.70036 |
| [19] | Borisov, A.V., Mamaev, I. S., and Karavaev, Yu. L., On the Loss of Contact of the Euler Disk, Nonlinear Dynam., 2015, vol. 79, no. 4, pp. 2287-2294. |
| [20] | Borisov, A. V., Mamaev, I. S., and Kilin, A.A., Dynamics of Rolling Disk, Regul. Chaotic Dyn., 2003, vol. 8, no. 2, pp. 201-212. · Zbl 1112.37320 |
| [21] | Borisov, A.V., Mamaev, I. S., and Tsyganov, A.V., Nonholonomic Dynamics and Poisson Geometry, Russian Math. Surveys, 2014, vol. 69, no. 3, pp. 481-538; see also: Uspekhi Mat. Nauk, 2014, vol. 69, no. 3(417), pp. 87-144. · Zbl 1348.70038 |
| [22] | Chaplygin, S.A., On a Motion of a Heavy Body of Revolution on a Horizontal Plane, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 119-130. · Zbl 1058.70002 |
| [23] | Cushman, R. H. and Duistermaat, J. J., Nearly Flat Falling Motions of the Rolling Disk, Regul. Chaotic Dyn., 2006, vol. 11, no. 1, pp. 31-60. · Zbl 1133.70308 |
| [24] | Devaney, R. L., Collision Orbits in the Anisotropic Kepler Problem, Invent. Math., 1978, vol. 45, no. 3, pp. 221-251. · Zbl 0382.58015 |
| [25] | Dolgopyat, D., Lectures on Bouncing Balls: Lecture Notes for a Course in Murcia (2013); https://www.math.umd.edu/dolgop/BBNotes2.pdf |
| [26] | Eldering, J., Normally Hyperbolic Invariant Manifolds: The Noncompact Case, Atlantis Studies in Dynamical Systems, vol. 2, Paris: Atlantis Press, 2013. · Zbl 1303.37011 |
| [27] | Fedorov, Yu.N., On Disk Rolling on Absolutely Rough Surface, Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, 1987, no. 4, pp. 67-75 (Russian). |
| [28] | Gallop, E. G., On the Rise of a Spinning Top, Proc. Camb. Phylos. Soc., 1904, vol. 19, no. 3, pp. 356-373. · JFM 35.0734.01 |
| [29] | García-Naranjo, L.C., Marrero, J.C., Non-Existence of an Invariant Measure for a Homogeneous Ellipsoid Rolling on the Plane, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 372-379. · Zbl 1338.37033 |
| [30] | Hadamard, J., Sur les mouvements de roulement, Mémoires de la Société des sciences physiques et naturelles de Bordeaux, sér. 4, 1895, vol. 5, pp. 397-417. · JFM 26.0849.02 |
| [31] | Hirsch, M.W., Pugh, C.C., and Shub, M., Invariant Manifolds, Berlin: Springer, 1977. · Zbl 0355.58009 |
| [32] | Kazakov, A. O., Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 508-520. · Zbl 1417.37223 |
| [33] | Kilin, A.A. and Pivovarova, E.N., The Rolling Motion of a Truncated Ball without Slipping and Spinning on a Plane, Regul. Chaotic Dyn., 2017, vol. 22, no. 3, pp. 298-317. · Zbl 1410.70010 |
| [34] | Koiller, J. and Ehlers, K. M., Rubber Rolling over a Sphere, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 127-152. · Zbl 1229.37089 |
| [35] | Korteweg, D., No article title, Über eine ziemlich verbrietete unrichtige Behandlungswiese eines Problemes der rolleden Bewegung und insbesondere über kleine rollende Schwingungen um eine Gleichgewichtslage, Nieuw Archief voor Wiskunde, 4, 130-155 (1899) · JFM 30.0639.02 |
| [36] | Kozlov, V.V., InvariantMeasures of Smooth Dynamical Systems, Generalized Functions and Summation Methods, Russian Acad. Sci. Izv. Math., 2016, vol. 80, no. 2, pp. 342-358; see also: Izv. Ross. Akad. Nauk. Ser. Mat., 2016, vol.80, no. 2, pp. 63-80. · Zbl 1417.37088 |
| [37] | Kozlov, V.V., Motion of a Disk on an Inclined Plane, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 1996, no. 5, pp. 29-35 (Russian). |
| [38] | Kozlov, V.V. and Kolesnikov, N.N., On Theorems of Dynamics, J. Appl. Math. Mech., 1978, vol. 42, no. 1, pp. 26-31; see also: Prikl. Mat. Mekh., 1978, vol. 42, no. 1, pp. 28-33. · Zbl 0418.70010 |
| [39] | Kozlov, V.V., Several Problems on Dynamical Systems and Mechanics, Nonlinearity, 2008, vol. 21, no. 9, pp. 149-155. · Zbl 1160.37026 |
| [40] | Littlewood, J.E., Some Problems in Real and Complex Analysis, Lexington,Mass.: D.C.Heath and Co., Raytheon Education Co., 1968. · Zbl 0185.11502 |
| [41] | Moser, J., Regularization of Kepler’s problem and the averaging method on a manifold, Comm. Pure Appl. Math., 1970, vol. 23, no. 4, pp. 609-636. · Zbl 0193.53803 |
| [42] | O’Reilly, O.M., The Dynamics of Rolling Disks and Sliding Disks, Nonlinear Dynam., 1996, vol. 10, no. 3, pp. 287-305. |
| [43] | Saari, D. G., Improbability of Collisions in Newtonian Gravitational Systems, Trans. Amer. Math. Soc., 1971, vol. 162, pp. 267-271. · Zbl 0231.70007 |
| [44] | Saari, D. G., Improbability of Collisions in Newtonian Gravitational Systems: 2, Trans. Amer. Math. Soc., 1973, vol. 181, pp. 351-368. · Zbl 0283.70007 |
| [45] | Stewart, D. E., Dynamics with Inequalities: Impacts and Hard Constraints, Philadelphia,Pa.: SIAM, 2011. · Zbl 1241.37003 |
| [46] | Vierkandt, A., Über gleitende und rollende Bewegung: 3. Das Rollen und Gleiten einer ebenen Fläche, insbesondere einer homogenen Kreisscheibe, auf der Horizontalebene unter dem Einfluss der Schwere, Monatsh. Math. Phys., 1892, vol. 3, no. 1, pp. 117-134. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
