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Regul. Chaotic Dyn., 2015, Volume 20, Issue 5, Pages 553–604 (Mi rcd21)  

This article is cited in 34 scientific papers (total in 34 papers)

Symmetries and Reduction in Nonholonomic Mechanics

Alexey V. Borisova, Ivan S. Mamaevb

a Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
b M. T. Kalashnikov Izhevsk State Technical University ul. Studencheskaya 7, Izhevsk, 426069 Russia

Abstract: This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.

Keywords: reduction, symmetry, tensor invariant, first integral, symmetry group, symmetry field, nonholonomic constraint, Noether theorem

Funding Agency Grant Number
Russian Science Foundation 15-12-20035
Sections 2-4 were written by A. V. Borisov and supported by the RSF grant no. 15-12-20035. Sections 5, 6 and Appendices were written by I. S. Mamaev within the framework of the RSF grant no. 14-19-01303.


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MSC: 37J60, 37J35, 37C10
Received: 01.08.2015

Citation: Alexey V. Borisov, Ivan S. Mamaev, “Symmetries and Reduction in Nonholonomic Mechanics”, Regul. Chaotic Dyn., 20:5 (2015), 553–604

Citation in format AMSBIB
\by Alexey V. Borisov, Ivan S. Mamaev
\paper Symmetries and Reduction in Nonholonomic Mechanics
\jour Regul. Chaotic Dyn.
\yr 2015
\vol 20
\issue 5
\pages 553--604

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    This publication is cited in the following articles:
    1. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Hamiltonization of elementary nonholonomic systems”, Russ. J. Math. Phys., 22:4 (2015), 444–453  crossref  mathscinet  zmath  isi  scopus
    2. 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  mathnet  crossref  mathscinet  adsnasa
    3. 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
    4. Alexey V. Borisov, Ivan S. Mamaev, “Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 232–248  mathnet  crossref  mathscinet  zmath  elib
    5. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “The Hess–Appelrot system and its nonholonomic analogs”, Proc. Steklov Inst. Math., 294 (2016), 252–275  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    6. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Generalizations of the Kovalevskaya case and quaternions”, Proc. Steklov Inst. Math., 295 (2016), 33–44  mathnet  crossref  crossref  mathscinet  isi  elib
    7. Yury A. Grigoryev, Alexey P. Sozonov, Andrey V. Tsiganov, “Integrability of Nonholonomic Heisenberg Type Systems”, SIGMA, 12 (2016), 112, 14 pp.  mathnet  crossref
    8. Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity”, Regul. Chaotic Dyn., 21:5 (2016), 556–580  mathnet  crossref  mathscinet  zmath  elib
    9. Y. Zhang, X.-S. Zhou, “Noether theorem and its inverse for nonlinear dynamical systems with nonstandard Lagrangians”, Nonlinear Dyn., 84:4 (2016), 1867–1876  crossref  mathscinet  zmath  isi  scopus
    10. 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
    11. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “O probleme Adamara–Gamelya i dinamike kolesnykh ekipazhei”, Nelineinaya dinam., 12:1 (2016), 145–163  mathnet
    12. Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period”, Regul. Chaotic Dyn., 21:4 (2016), 455–476  mathnet  crossref
    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. I. A. Bizyaev, A. V. Borisov, S. P. Kuznetsov, “Chaplygin sleigh with periodically oscillating internal mass”, EPL, 119:6 (2017), 60008  crossref  isi
    15. 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  mathnet  crossref  mathscinet  isi  scopus
    16. S. P. Kuznetsov, “Regular and chaotic motions of the Chaplygin sleigh with periodically switched location of nonholonomic constraint”, EPL, 118:1 (2017), 10007  crossref  mathscinet  isi  scopus
    17. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Sluchai Gessa–Appelrota i kvantovanie chisla vrascheniya”, Nelineinaya dinam., 13:3 (2017), 433–452  mathnet  crossref  mathscinet  elib
    18. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Hess–Appelrot Case and Quantization of the Rotation Number”, Regul. Chaotic Dyn., 22:2 (2017), 180–196  mathnet  crossref  mathscinet  zmath
    19. L. C. Garcia-Naranjo, J. Montaldi, “Gauge momenta as Casimir functions of nonholonomic systems”, Arch. Ration. Mech. Anal., 228:2 (2018), 563–602  crossref  mathscinet  zmath  isi  scopus
    20. 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
    21. 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  mathnet  crossref
    22. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Dynamics of the Chaplygin ball on a rotating plane”, Russ. J. Math. Phys., 25:4 (2018), 423–433  crossref  mathscinet  zmath  isi  scopus
    23. B. Jovanovic, “Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems”, J. Geom. Mech., 10:2 (2018), 173–187  crossref  mathscinet  zmath  isi  scopus
    24. Andrey A. Ardentov, Yury L. Karavaev, Kirill S. Yefremov, “Euler Elasticas for Optimal Control of the Motion of Mobile Wheeled Robots: the Problem of Experimental Realization”, Regul. Chaotic Dyn., 24:3 (2019), 312–328  mathnet  crossref
    25. 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
    26. 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
    27. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Dynamics of a Chaplygin sleigh with an unbalanced rotor: regular and chaotic motions”, Nonlinear Dyn., 98:3 (2019), 2277–2291  crossref  mathscinet  zmath  isi  scopus
    28. L. C. Garcia-Naranjo, “Generalisation of Chaplygin's reducing multiplier theorem with an application to multi-dimensional nonholonomic dynamics”, J. Phys. A-Math. Theor., 52:20 (2019), 205203  crossref  mathscinet  isi  scopus
    29. Ch. Liu, L. Dong, “Physics-based control education: energy, dissipation, and structure assignments”, Eur. J. Phys., 40:3 (2019), 035006  crossref  isi  scopus
    30. A. Borisov, A. Kilin, I. Mamaev, “Invariant submanifolds of genus 5 and a Cantor staircase in the nonholonomic model of a snakeboard”, Int. J. Bifurcation Chaos, 29:3 (2019), 1930008  crossref  mathscinet  zmath  isi  scopus
    31. A. Borisov, I. Mamaev, “Rigid body dynamics”, Rigid Body Dynamics, de Gruyter Studies in Mathematical Physics, 52, Walter de Gruyter Gmbh, 2019, 1–520  mathscinet  isi
    32. V. Gzenda, V. Putkaradze, “Integrability and chaos in figure skating”, J. Nonlinear Sci., 30:3 (2020), 831–850  crossref  mathscinet  zmath  isi  scopus
    33. Elizaveta M. Artemova, Yury L. Karavaev, Ivan S. Mamaev, Evgeny V. Vetchanin, “Dynamics of a Spherical Robot with Variable Moments of Inertia and a Displaced Center of Mass”, Regul. Chaotic Dyn., 25:6 (2020), 689–706  mathnet  crossref  mathscinet
    34. Alexander A. Kilin, Elena N. Pivovarova, “Stability and Stabilization of Steady Rotations of a Spherical Robot on a Vibrating Base”, Regul. Chaotic Dyn., 25:6 (2020), 729–752  mathnet  crossref  mathscinet
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