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Regul. Chaotic Dyn., 2015, Volume 20, Issue 3, Pages 383–400 (Mi rcd2)  

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

The Jacobi Integral in Nonholonomic Mechanics

A. V. Borisovab, I. S. Mamaevac, I. A. Bizyaevad

a Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034, Russia
b Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700, Russia
c M. T. Kalashnikov Izhevsk State Technical University, ul. Studencheskaya 7, Izhevsk, 426069, Russia
d Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

Abstract: In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.

Keywords: nonholonomic constraint, Jacobi integral, Chaplygin sleigh, rotating table, Suslov problem.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work was supported by the Russian Scientific Foundation (project 14-50-00005) and was carried out at the Steklov Mathematical Institute of the Russian Academy of Sciences.


DOI: https://doi.org/10.1134/S1560354715030107

References: PDF file   HTML file

Bibliographic databases:

MSC: 70F25, 37J60, 70E18
Received: 28.04.2015
Accepted:13.05.2015
Language:

Citation: A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “The Jacobi Integral in Nonholonomic Mechanics”, Regul. Chaotic Dyn., 20:3 (2015), 383–400

Citation in format AMSBIB
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\paper The Jacobi Integral in Nonholonomic Mechanics
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\pages 383--400
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    Citing articles on Google Scholar: Russian citations, English citations
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    Translation

    This publication is cited in the following articles:
    1. Francesco Fassò, Nicola Sansonetto, “Conservation of Energy and Momenta in Nonholonomic Systems with Affine Constraints”, Regul. Chaotic Dyn., 20:4 (2015), 449–462  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    2. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Hamiltonization of elementary nonholonomic systems”, Russian Journal of Mathematical Physics, 22:4 (2015), 444–453  crossref  mathscinet  zmath  scopus
    3. A. V. Borisov, I. S. Mamaev, “Equations of motion of non-holonomic systems”, Russian Math. Surveys, 70:6 (2015), 1167–1169  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Ivan A. Bizyaev, Alexey V. Borisov, Alexey O. Kazakov, “Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors”, Regul. Chaotic Dyn., 20:5 (2015), 605–626  mathnet  crossref  mathscinet  zmath  elib
    5. Alexey V. Borisov, Yury L. Karavaev, Ivan S. Mamaev, Nadezhda N. Erdakova, Tatyana B. Ivanova, Valery V. Tarasov, “Experimental Investigation of the Motion of a Body with an Axisymmetric Base Sliding on a Rough Plane”, Regul. Chaotic Dyn., 20:5 (2015), 518–541  mathnet  crossref  mathscinet  zmath
    6. Alexey V. Borisov, Ivan S. Mamaev, “Symmetries and Reduction in Nonholonomic Mechanics”, Regul. Chaotic Dyn., 20:5 (2015), 553–604  mathnet  crossref  mathscinet  zmath
    7. V. Kozlov, “The phenomenon of reversal in the Euler–Poincaré––Suslov nonholonomic systems”, J. Dyn. Control Syst., 2016 (to appear)  mathnet  crossref  mathscinet  scopus
    8. 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
    9. Yu. L. Karavaev, A. A. Kilin, “Nonholonomic dynamics and control of a spherical robot with an internal omniwheel platform: theory and experiments”, Proc. Steklov Inst. Math., 295 (2016), 158–167  mathnet  crossref  crossref  mathscinet  isi  elib
    10. 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
    11. 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
    12. A. P. Ivanov, “The ANAIS billiard experiment”, Dokl. Phys., 61:6 (2016), 285–287  crossref  mathscinet  isi  scopus
    13. F. Fasso, N. Sansonetto, “Conservation of ‘moving’ energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces”, J. Nonlinear Sci., 26:2 (2016), 519–544  crossref  mathscinet  zmath  isi  scopus
    14. Alexey V. Borisov, Ivan S. Mamaev, “An Inhomogeneous Chaplygin Sleigh”, Regul. Chaotic Dyn., 22:4 (2017), 435–447  mathnet  crossref  mathscinet
    15. 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
    16. 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  mathnet  crossref
    17. K. Zengel, “The electromagnetic analogy of a ball on a rotating conical turntable”, Am. J. Phys., 85:12 (2017), 901–907  crossref  isi  scopus
    18. A. V. Borisov, I. S. Mamaev, “Neodnorodnye sani Chaplygina”, Nelineinaya dinam., 13:4 (2017), 625–639  mathnet  crossref  elib
    19. V. Fedonyuk, Ph. Tallapragada, “Sinusoidal control and limit cycle analysis of the dissipative Chaplygin sleigh”, Nonlinear Dyn., 93:2 (2018), 835–846  crossref  isi  scopus
    20. A. V. Borisov, I. S. Mamaev, A. A. Kilin, “A Nonholonomic Model of the Paul Trap”, Regul. Chaotic Dyn., 23:3 (2018), 339–354  mathnet  crossref  mathscinet  adsnasa
    21. F. Fasso, L. C. Garcia-Naranjo, N. Sansonetto, “Moving energies as first integrals of nonholonomic systems with affine constraints”, Nonlinearity, 31:3 (2018), 755–782  crossref  mathscinet  zmath  isi  scopus
    22. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane”, Regul. Chaotic Dyn., 23:6 (2018), 665–684  mathnet  crossref  mathscinet
    23. A. A. Kilin, E. N. Pivovarova, “Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges”, Regul. Chaotic Dyn., 23:7-8 (2018), 887–907  mathnet  crossref
    24. 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
    25. 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
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