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Regul. Chaotic Dyn., 2002, Volume 7, Issue 2, Pages 161–176 (Mi rcd810)  

This article is cited in 24 scientific papers (total in 25 papers)

Nonholonomic Systems

On the integration theory of equations of nonholonomic mechanics

V. V. Kozlov

Department of Mechanics and Mathematics, Moscow State University, Vorob'ievy Gory, 119899, Moscow, Russia

Abstract: The paper deals with the problem of integration of equations of motion in nonholonomic systems. By means of well-known theory of the differential equations with an invariant measure the new integrable systems are discovered. Among them there are the generalization of Chaplygin's problem of rolling nonsymmetric ball in the plane and the Suslov problem of rotation of rigid body with a fixed point. The structure of dynamics of systems on the invariant manifold in the integrable problems is shown. Some new ideas in the theory of integration of the equations in nonholonomic mechanics are suggested. The first of them consists in using known integrals as the constraints. The second is the use of resolvable groups of symmetries in nonholonomic systems. The existence conditions of invariant measure with analytical density for the differential equations of nonholonomic mechanics is given.


Bibliographic databases:

MSC: 37J60, 37J35

Citation: V. V. Kozlov, “On the integration theory of equations of nonholonomic mechanics”, Regul. Chaotic Dyn., 7:2 (2002), 161–176

Citation in format AMSBIB
\by V. V. Kozlov
\paper On the integration theory of equations of nonholonomic mechanics
\jour Regul. Chaotic Dyn.
\yr 2002
\vol 7
\issue 2
\pages 161--176

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    This publication is cited in the following articles:
    1. I. S. Mamaev, “Universalnyi kompleks programm dlya issledovaniya mekhanicheskikh sistem s negolonomnymi svyazyami”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2009, no. 2, 147–160  mathnet
    2. Peter Lynch, Miguel D. Bustamante, “Quaternion Solution for the Rock’n’roller: Box Orbits, Loop Orbits and Recession”, Regul. Chaotic Dyn., 18:1-2 (2013), 166–183  mathnet  crossref  mathscinet  zmath
    3. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Geometrizatsiya teoremy Chaplygina o privodyaschem mnozhitele”, Nelineinaya dinam., 9:4 (2013), 627–640  mathnet
    4. A. V. Tsyganov, “O share Chaplygina v absolyutnom prostranstve”, Nelineinaya dinam., 9:4 (2013), 711–719  mathnet
    5. Andrey V. Tsiganov, “On the Lie Integrability Theorem for the Chaplygin Ball”, Regul. Chaotic Dyn., 19:2 (2014), 185–197  mathnet  crossref  mathscinet  zmath
    6. Andrey V. Tsiganov, “On Integrable Perturbations of Some Nonholonomic Systems”, SIGMA, 11 (2015), 085, 19 pp.  mathnet  crossref
    7. V. V. Kozlov, “Dinamika sistem s servosvyazyami. I”, Nelineinaya dinam., 11:2 (2015), 353–376  mathnet  elib
    8. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Integral Yakobi v negolonomnoi mekhanike”, Nelineinaya dinam., 11:2 (2015), 377–396  mathnet
    9. A. V. Borisov, Yu. L. Karavaev, I. S. Mamaev, N. N. Erdakova, T. B. Ivanova, V. V. Tarasov, “Eksperimentalnoe issledovanie dvizheniya tela s osesimmetrichnym osnovaniem, skolzyaschego po sherokhovatoi ploskosti”, Nelineinaya dinam., 11:3 (2015), 547–577  mathnet
    10. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Hojman Construction and Hamiltonization of Nonholonomic Systems”, SIGMA, 12 (2016), 012, 19 pp.  mathnet  crossref
    11. Alexey V. Borisov, Alexey O. Kazakov, Igor R. Sataev, “Spiral Chaos in the Nonholonomic Model of a Chaplygin Top”, Regul. Chaotic Dyn., 21:7-8 (2016), 939–954  mathnet  crossref
    12. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Istoriko-kriticheskii obzor razvitiya negolonomnoi mekhaniki: klassicheskii period”, Nelineinaya dinam., 12:3 (2016), 385–411  mathnet  crossref  zmath  elib
    13. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    14. Sergey P. Kuznetsov, “Regular and Chaotic Dynamics of a Chaplygin Sleigh due to Periodic Switch of the Nonholonomic Constraint”, Regul. Chaotic Dyn., 23:2 (2018), 178–192  mathnet  crossref
    15. Shengda Hu, Manuele Santoprete, “Suslov Problem with the Clebsch–Tisserand Potential”, Regul. Chaotic Dyn., 23:2 (2018), 193–211  mathnet  crossref
    16. S. P. Kuznetsov, “Complex Dynamics in Generalizations of the Chaplygin Sleigh”, Rus. J. Nonlin. Dyn., 15:4 (2019), 551–559  mathnet  crossref  elib
    17. Miguel D. Bustamante, Peter Lynch, “Nonholonomic Noetherian Symmetries and Integrals of the Routh Sphere and the Chaplygin Ball”, Regul. Chaotic Dyn., 24:5 (2019), 511–524  mathnet  crossref  mathscinet
    18. Ogul Esen, Victor M. Jiménez, Manuel de León, Cristina Sardón, “Reduction of a Hamilton – Jacobi Equation for Nonholonomic Systems”, Regul. Chaotic Dyn., 24:5 (2019), 525–559  mathnet  crossref  mathscinet
    19. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem”, Regul. Chaotic Dyn., 24:5 (2019), 560–582  mathnet  crossref  mathscinet
    20. Liu Ch. Dong L., “Physics-Based Control Education: Energy, Dissipation, and Structure Assignments”, Eur. J. Phys., 40:3 (2019), 035006  crossref  isi  scopus
    21. Borisov A., Mamaev I., “Rigid Body Dynamics”, Rigid Body Dynamics, de Gruyter Studies in Mathematical Physics, 52, Walter de Gruyter Gmbh, 2019, 1–520  mathscinet  isi
    22. S. V. Gonchenko, A. S. Gonchenko, A. O. Kazakov, “Three Types of Attractors and Mixed Dynamics of Nonholonomic Models of Rigid Body Motion”, Proc. Steklov Inst. Math., 308 (2020), 125–140  mathnet  crossref  crossref  mathscinet  isi  elib
    23. Gzenda V. Putkaradze V., “Integrability and Chaos in Figure Skating”, J. Nonlinear Sci., 30:3 (2020), 831–850  crossref  mathscinet  zmath  isi  scopus
    24. Vladimir Dragović, Borislav Gajić, Božidar Jovanović, “Demchenko's nonholonomic case of a gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis”, Theor. Appl. Mech., 47:2 (2020), 257–287  mathnet  crossref
    25. Szuminski W. Przybylska M., “Differential Galois Integrability Obstructions For Nonlinear Three-Dimensional Differential Systems”, Chaos, 30:1 (2020), 013135  crossref  mathscinet  zmath  isi  scopus
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