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Regular and Chaotic Dynamics, 2015, Volume 20, Issue 3, Pages 383–400
DOI: https://doi.org/10.1134/S1560354715030107 (Mi rcd2) 		 
This article is cited in 44 scientific papers (total in 44 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
Citations (44)
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DOI: https://doi.org/10.1134/S1560354715030107
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.
Received: 28.04.2015
Accepted: 13.05.2015
Bibliographic databases:
Document Type: Article
MSC: 70F25, 37J60, 70E18
Language: English
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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\pages 383--400
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    Translation
    This publication is cited in the following 44 articles:
    1. Mariana Costa-Villegas, Luis C. García-Naranjo, “Affine Generalizations of the Nonholonomic Problem of a Convex Body Rolling without Slipping on the Plane”, Regul. Chaot. Dyn., 2025  crossref
    2. A. G. Agúndez, D. García-Vallejo, E. Freire, “Analytical and numerical stability analysis of a toroidal wheel with nonholonomic constraints”, Nonlinear Dyn, 112:4 (2024), 2453  crossref
    3. Luis C. García-Naranjo, Juan C. Marrero, David Martín de Diego, Paolo E. Petit Valdés, “Almost-Poisson Brackets for Nonholonomic Systems with Gyroscopic Terms and Hamiltonisation”, J Nonlinear Sci, 34:6 (2024)  crossref
    4. Alexander A. Kilin, Elena N. Pivovarova, Tatiana B. Ivanova, “Rolling of a Homogeneous Ball on a Moving Cylinder”, Regul. Chaot. Dyn., 2024  crossref
    5. Francesco Fassò, Nicola Sansonetto, “On Some Aspects of the Dynamics of a Ball in a Rotating Surface of Revolution and of the Kasamawashi Art”, Regul. Chaotic Dyn., 27:4 (2022), 409–423  mathnet  crossref  mathscinet
    6. Marco Dalla Via, Francesco Fassò, Nicola Sansonetto, “On the Dynamics of a Heavy Symmetric Ball that Rolls Without Sliding on a Uniformly Rotating Surface of Revolution”, J Nonlinear Sci, 32:6 (2022)  crossref
    7. Alexander A. Kilin, Elena N. Pivovarova, “A Particular Integrable Case in the Nonautonomous Problem of a Chaplygin Sphere Rolling on a Vibrating Plane”, Regul. Chaotic Dyn., 26:6 (2021), 775–786  mathnet  crossref
    8. T. B. Ivanova, “Non-holonomic rolling of a ball on the surface of a rotating cylinder”, ZAMM-Z. Angew. Math. Mech., 100:12 (2020), e202000067  crossref  mathscinet  isi  scopus
    9. A. V. Borisov, T. B. Ivanova, A. A. Kilin, I. S. Mamaev, “Circular orbits of a ball on a rotating conical turntable”, Acta Mech., 231:3 (2020), 1021–1028  crossref  mathscinet  zmath  isi  scopus
    10. Yu. L. Karavaev, A. A. Kilin, “The Dynamics of a Spherical Robot of Combined Type by Periodic Control Actions”, Rus. J. Nonlin. Dyn., 15:4 (2019), 497–504  mathnet  crossref  mathscinet  elib
    11. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness”, Regul. Chaotic Dyn., 24:3 (2019), 329–352  mathnet  crossref
    12. 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
    13. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Comment on “Confining rigid balls by mimicking quadrupole ion trapping” [Am. J. Phys. 85, 821 (2017)]”, Am. J. Phys., 87:11 (2019), 935–938  crossref  isi  scopus
    14. I. A. Bizyaev, A. V. Borisov, V. V. Kozlov, I. S. Mamaev, “Fermi-like acceleration and power-law energy growth in nonholonomic systems”, Nonlinearity, 32:9 (2019), 3209–3233  crossref  mathscinet  zmath  isi  scopus
    15. A. V. Borisov, T. B. Ivanova, A. A. Kilin, I. S. Mamaev, “Nonholonomic rolling of a ball on the surface of a rotating cone”, Nonlinear Dyn., 97:2 (2019), 1635–1648  crossref  zmath  isi  scopus
    16. 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
    17. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “A Nonholonomic Model of the Paul Trap”, Regul. Chaotic Dyn., 23:3 (2018), 339–354  mathnet  crossref  mathscinet  adsnasa
    18. 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
    19. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “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
    20. Alexander A. Kilin, Elena N. Pivovarova, “Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges”, Regul. Chaotic Dyn., 23:7-8 (2018), 887–907  mathnet  crossref  mathscinet
    Citing articles in Google Scholar: Russian citations, English citations
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