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 Regul. Chaotic Dyn., 2015, Volume 20, Issue 6, Pages 752–766 (Mi rcd34)

On the Hadamard–Hamel Problem and the Dynamics of Wheeled Vehicles

Alexey V. Borisova, Alexander A. Kilinb, Ivan S. Mamaevc

a Moscow Institute of Physics and Technology, Dolgoprudnyi, 141700, Russia
b Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034, Russia
c Steklov Mathematical Institute, ul. Gubkina 8, Moscow, 119991, Russia

Abstract: In this paper, we develop the results obtained by J.Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs.

Keywords: nonholonomic constraint, wheeled vehicle, reduction, equations of motion

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation Russian Foundation for Basic Research 15-38-20879 mol_a_ved13-01-12462-ofi_m The work of A.V.Borisov was carried out within the framework of the state assignment for institutions of higher education. The work of A.A.Kilin was supported by the RFBR grant no. 15-38-20879 mol_a_ved. The work of I.S.Mamaev was supported by the RFBR grant no. 13-01-12462-ofi_m.

DOI: https://doi.org/10.1134/S1560354715060106

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Document Type: Article
MSC: 37J60, 37N05
Language: English

Citation: Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “On the Hadamard–Hamel Problem and the Dynamics of Wheeled Vehicles”, Regul. Chaotic Dyn., 20:6 (2015), 752–766

Citation in format AMSBIB
\Bibitem{BorKilMam15} \by Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev \paper On the Hadamard--Hamel Problem and the Dynamics of Wheeled Vehicles \jour Regul. Chaotic Dyn. \yr 2015 \vol 20 \issue 6 \pages 752--766 \mathnet{http://mi.mathnet.ru/rcd34} \crossref{https://doi.org/10.1134/S1560354715060106} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3431189} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2015RCD....20..752B} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000365809000010} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84948952940} 

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This publication is cited in the following articles:
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4. T. B. Ivanova, N. N. Erdakova, Yu. L. Karavaev, “Experimental investigation of the dynamics of a brake shoe”, Dokl. Phys., 61:12 (2016), 611–614
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6. A. V. Borisov, I. S. Mamaev, “Neodnorodnye sani Chaplygina”, Nelineinaya dinam., 13:4 (2017), 625–639
7. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics”, Russian Math. Surveys, 72:5 (2017), 783–840
8. I. A. Bizyaev, “Sani Chaplygina s dvizhuscheisya tochechnoi massoi”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:4 (2017), 583–589
9. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration”, Regul. Chaotic Dyn., 22:8 (2017), 955–975
10. I. A. Bizyaev, A. V. Borisov, S. P. Kuznetsov, “Chaplygin sleigh with periodically oscillating internal mass”, EPL, 119:6 (2017), 60008
11. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Hamilton's Principle and the Rolling Motion of a Symmetric Ball”, Dokl. Phys., 62:6 (2017), 314–317
12. Alexey V. Borisov, Ivan S. Mamaev, “An Inhomogeneous Chaplygin Sleigh”, Regul. Chaotic Dyn., 22:4 (2017), 435–447
13. O. N. Kirillov, “Locating the sets of exceptional points in dissipative systems and the self-stability of bicycles”, Entropy, 20:7 (2018), 502
14. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Exotic Dynamics of Nonholonomic Roller Racer with Periodic Control”, Regul. Chaotic Dyn., 23:7-8 (2018), 983–994