Abstract:
The classical question whether nonholonomic dynamics is realized as limit of friction forces was first posed by Carathéodory. It is known that, indeed, when friction forces are scaled to infinity, then nonholonomic dynamics is obtained as a singular limit.
Our results are twofold. First, we formulate the problem in a differential geometric context. Using modern geometric singular perturbation theory in our proof, we then obtain a sharp statement on the convergence of solutions on infinite time intervals. Secondly, we set up an explicit scheme to approximate systems with large friction by a perturbation of the nonholonomic dynamics. The theory is illustrated in detail by studying analytically and numerically the Chaplygin sleigh as an example. This approximation scheme offers a reduction in dimension and has potential use in applications.
This publication is cited in the following 15 articles:
Alexander Koshelev, Eugene Kugushev, Tatiana Shahova, Springer Proceedings in Mathematics & Statistics, 453, Perspectives in Dynamical Systems I — Applications, 2024, 319
A. A. Koshelev, E. I. Kugushev, T. V. Shahova, “On the motion of a ball between rotating planes with viscous friction”, Moscow University Mеchanics Bulletin, 79:3 (2024), 110–117
Vaughn Gzenda, Robin Chhabra, “Affine connection approach to the realization of nonholonomic constraints by strong friction forces”, Nonlinear Dyn, 2024
E. V. Vetchanin, “The Motion of a Balanced Circular Cylinder in an Ideal Fluid Under the Action of External Periodic Force and Torque”, Rus. J. Nonlin. Dyn., 15:1 (2019), 41–57
T. B. Ivanova, “The Rolling of a Homogeneous Ball with Slipping on a Horizontal Rotating Plane”, Rus. J. Nonlin. Dyn., 15:2 (2019), 171–178
M. D. Kvalheim, B. Bittner, Sh. Revzen, “Gait modeling and optimization for the perturbed Stokes regime”, Nonlinear Dyn., 97:4 (2019), 2249–2270
Alexander Kobrin, Vladimir Sobolev, Trends in Mathematics, 11, Extended Abstracts Spring 2018, 2019, 1
Alexey V. Borisov, Ivan S. Mamaev, Eugeny V. Vetchanin, “Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation”, Regul. Chaotic Dyn., 23:4 (2018), 480–502
Alexey V. Borisov, Sergey P. Kuznetsov, “Comparing Dynamics Initiated by an Attached Oscillating Particle for the Nonholonomic Model of a Chaplygin Sleigh and for a Model with Strong Transverse and Weak Longitudinal Viscous Friction Applied at a Fixed Point on the Body”, Regul. Chaotic Dyn., 23:7-8 (2018), 803–820
J. Eldering, M. Kvalheim, Sh. Revzen, “Global linearization and fiber bundle structure of invariant manifolds”, Nonlinearity, 31:9 (2018), 4202–4245
A Kobrin, V Sobolev, “Decomposition of nonholonomic mechanics models”, J. Phys.: Conf. Ser., 1096 (2018), 012054
S. Koshkin, V. Jovanovic, “Realization of non-holonomic constraints and singular perturbation theory for plane dumbbells”, J. Eng. Math., 106:1 (2017), 123–141
A. Kobrin, V. Sobolev, “Integral manifolds of fast-slow systems in nonholonomic mechanics”, 3rd International Conference Information Technology and Nanotechnology (ITNT-2017), Procedia Engineering, 201, eds. V. Soifer, N. Kazanskiy, O. Korotkova, S. Sazhin, Elsevier Science BV, 2017, 556–560
Alexander P. Ivanov, “On Final Motions of a Chaplygin Ball on a Rough Plane”, Regul. Chaotic Dyn., 21:7-8 (2016), 804–810
Citation in format AMSBIB
Contact us: