Qualitative Theory of Dynamical Systems

, Volume 4, Issue 2, pp 383–411 | Cite as

A note on the conservation of energy and volume in the setting of nonholonomic mechanical systems

  • Marcelo H. Kobayashi
  • Waldyr M. Oliva


This note concerns the analysis of conservation of energy and volume for a series of well known examples of nonholonomic mechanical systems, with linear and non-linear constraints, and aims to make evident some geometric aspects related with them.

Key Words

mechanical systems energy and volume conservation nonholonomic constraints 


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  1. 1.
    P. Appell,Sur les liasons exprimées par des relations non linéaires entre les vitesses, Comptes Rendus de l'Acadmie des Sciences Paris152 (1911), 1197–1199.MATHGoogle Scholar
  2. 2.
    V. I. Arnold,Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics60, Springer-Verlag, 2nd ed., 1989.Google Scholar
  3. 3.
    S. Benenti,Geometrical Aspects of the Dynamics of Non-Holonomic Systems, Rendiconti del Seminario Matematico dell' Univivesitá e del Politecnico di Torino54 (1996), 203–212.MATHMathSciNetGoogle Scholar
  4. 4.
    L. O. Biscolla,Uma generalização do Problema de Kendall usando Métodos da Teoria Geométrica de Controle, ongoing work — São Paulo, 2003.Google Scholar
  5. 5.
    C. J. Blackall,On volume integral invariants of non-holonomic dynamical systems, American Journal of Mathematics,63:1 (1941), 155–168.CrossRefMathSciNetGoogle Scholar
  6. 6.
    A.M. Bloch, P.S. Krishnaprasad, J.E. Marsden andR. Murray,Nonholonomic Mechanical Systems and Symmetry, Archive for Rational Mechanics and Analysis136 (1996), 21–99.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    E. Cartan,Sur la Représentation Géométrique des Sistèmes Matériels Nonholohomes, Proceedings of the International Congress of Mathematicians4 (1928), 253–261.MathSciNetGoogle Scholar
  8. 8.
    H. Cendra, J. E. Marsden and T. S. Ratiu,Geometric Mechanics Lagrangian Reduction, and Nonholonomic Systems, in Mathematics Unlimited-2001 and Beyond, B. Engquist and W. Schmid eds., Springer-Verlag, 2000.Google Scholar
  9. 9.
    S. A. Chaplygin,On a ball's rolling on a horizontal plane, Reg. Chaot. Dyn.,7:2, (2000), 131–148. Original paper in Math. Sbornik24, 139–168, (1903).CrossRefMathSciNetGoogle Scholar
  10. 10.
    P. Dazord,Mécanique Hamiltonienne en Présence de Constraintes, Illinois Journal of Mathematics38 (1994), 148–175.MATHMathSciNetGoogle Scholar
  11. 11.
    Yu. N. Fedorov and V. V. Kozlov,Various Aspects of n-Dimensional Rigid Body Dynamics in Kozlov, V.V. (editor) Dynamical Systems in Classical Mechanics, volume 168 ofAMS Translations series 2, 1995.Google Scholar
  12. 12.
    G. Fusco andW. M. Oliva,Dissipative Systems with Constraints, Journal of Differential Equations63 (1986), 362–388.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    G. Gallavotti andD. Ruelle, SRBStates and Nonequilibrium Statistical Mechanics Close to Equilibrium, Communication in Mathematical Physics190 (1997), 279–285.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    G. Hamel,Theoretische Mechanik: Eine einheitliche Einfhrung in die gesamte Mechanik volume 57 ofGrundlehren der Mathematischen Wissenschaften, 1949; revised edition, Springer-Verlag, Berlin-New York, 1978.Google Scholar
  15. 15.
    S. Helgason,Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics80, 1978.Google Scholar
  16. 16.
    W.G. Hoover,Molecular Dynamics, Lecture Notes in Physics258, Springer-Verlag, 1986.Google Scholar
  17. 17.
    M. H. Kobayashi andW. M. Oliva,On the Birkhoff Approach to Classical Mechanics, Resenhas IME-USP6 (2003), 1–71.MathSciNetGoogle Scholar
  18. 18.
    J. Koiller,Reduction of Some Classical Non-Holonomic Systems with Symmetry, Archive for Rational Mechanics and Analysis118 (1992), 113–148.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    J. Koiller, P. R. Rodrigues andP. Pitanga,Non-holonomic Connections Following Éllie Cartan, Anais da Academia Brasileira de Ciências73 (2001), 165–190.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    I. Kupka andW. M. Oliva,The Non-Holonomic Mechanics, Journal of Differential Equations169 (2001), 169–189.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    A. Lewis, J. P. Ostrowski, R. M. Murray and J. Burdick,Nonholonomic mechanics and locomotion: the snakeboard example, IEEE International Conference on Robotics and Automation, 1994.Google Scholar
  22. 22.
    C.-M. Marle,Reduction of Constrained Mechanical Systems and Stability of Relative Equilibria, Communications in Mathematical Physics174 (1995), 295–318.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    C.-M. Marle,Kinematic and Geometric Constraints, Servomechanisms and Control of Mechanical Systems, Rendiconti del Seminario Matematico dell' Univivesitá e del Politecnico di Torino54 (1996), 353–364.MATHMathSciNetGoogle Scholar
  24. 24.
    J. I. Neimark and N. A. Fufaev,Dynamics of Nonholonomic Constraints, Translations of Mathematical Monographs33, American Mathematical Society, 1972.Google Scholar
  25. 25.
    W. M. Oliva,Geometric Mechanics, Lecture Notes in Mathematics1798, Springer-Verlag, 2002.Google Scholar
  26. 26.
    J. Ostrowski,Geometric Perspectives on the Mechanics and Control of Undulatory Locomotion, Ph.D. dissertation, California Institute of Technology, USA, 1995.Google Scholar
  27. 27.
    L. A. Pars,A Treatise on Analytical Dynamics, Heinemann Educational Books, 1965.Google Scholar
  28. 28.
    R.M. Rosenberg,Analytical Dynamics of Discrete Systems, Plenum Press, 1977.Google Scholar
  29. 29.
    D. Ruelle,Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics, Journal of Statistical Mechanics95 1999, 393–468.MATHMathSciNetGoogle Scholar
  30. 30.
    D. Schneider,Nonholonomic Euler-Poincare Equations and Stability in Chaplygin's Sphere, Dyn. Syst.: an Intern. J.,17: 2, (2002) 87–130.MATHCrossRefGoogle Scholar
  31. 31.
    J. N. Tavares,About Cartan Geometrization of Non Holonomic Mechanics, Journal of Geometry and Physics45 (2003), 1–23.MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    G. Terra andM.H. Kobayashi,On Classical Mechanics with Nonlinear Constraint, in press at Journal of Geometry and Physics49 (2003), 385–417.CrossRefMathSciNetGoogle Scholar
  33. 33.
    G. Terra andM.H. Kobayashi,On the Variational Mechanics with Non-Linear Constraints, Journal de Mathématiques Pures et Appliquées83 (2004), 629–671.MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    A. M. Vershik and V. Ya Gershkovich,Non-Holonomic Dynamical Systems, Geometry of Distributions and Variational Problems, in Encyclopaedia of Mathematical Sciences: Dynamical Systems VII,16, Springer-Verlag, 1994.Google Scholar
  35. 35.
    A. P. Veselov andL. E. Veselova,Flows on Lie groups with a nonholonomic constraint and integrable non-Hamiltonian systems (Russian), Funktsional. Anal. i Prilozhen.20:4, (1986), 65–66. English translation: Functional Anal. Appl.20: 4,308–309.MathSciNetGoogle Scholar
  36. 36.
    A. P. Veselov andL. E. Veselova,Integrable nonholonomic systems on Lie groups (Russian) Mat. Zametki44:5 (1988), 604–619, 701; English translation: Math. Notes44:5/6 (1989), 810–819.MATHMathSciNetGoogle Scholar
  37. 37.
    M. P. Wojtkowski,Magnetic Flows and Gaussian Thermostats on Manifolds of Negative Curvature, Fundamenta Mathematicae163 2000, 177–191.MATHMathSciNetGoogle Scholar
  38. 38.
    G. Zampieri,Dynamic Convexity for Natural Thermostatted Systems, to appear in Journal of Differential Equations.Google Scholar

Copyright information

© Birkhäuser-Verlag 2004

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of Hawai'i at ManoaHonolulu
  2. 2.Instituto Superior TécnicoISR and CAMGSDLisbonPortugal

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