SeMA Journal

, Volume 51, Issue 1, pp 141–148 | Cite as

An Inverse Problem on Vakonomic Mechanics

  • Waldyr M. Oliva
  • Gláucio Terra
Actas del NSDS09


We study a version of the inverse problem of Calculus of Variations in the context of Vakonomic Mechanics.

Key words

Vakonomic Mechanics Inverse Problem of the Calculus of Variations 

AMS subject classifications

49N45 37J60 


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  1. [1]
    I. M. Anderson, The Variational Bicomplex, To Appear.Google Scholar
  2. [2]
    V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Dynamical Systems III, vol. 3 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, New York, 1988, ch. Mathematical Aspects of Classical and Celestial Mechanics, pp. 1–286.Google Scholar
  3. [3]
    J. Douglas, Solution of the inverse problem of the calculus of variations, Proc. Nat. Acad. Sci. U.S.A., 25 (1939), pp. 631–637.MathSciNetCrossRefGoogle Scholar
  4. [4]
    —, Solution of the inverse problem of the calculus of variations, Trans. Amer. Math. Soc., 50 (1941), pp. 71–128.MathSciNetCrossRefGoogle Scholar
  5. [5]
    H. Helmholtz, Uber der physikalische bedeutung des princips der kleinsten wirkung, J. Reine Angew. Math., 100 (1887), pp. 137–166.Google Scholar
  6. [6]
    I. Kupka and W. M. Oliva, The non-holonomic mechanics, Journal of Differential Equations, 169 (2001), pp. 169–189.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    W. M. Oliva, Geometric Mechanics, vol. 1798 of Lecture Notes in Mathematics, Springer-Verlag, 2002.Google Scholar
  8. [8]
    W. M. Oliva and G. Terra, An inverse problem on vakonomic mechanics, Preprint to be Submitted.Google Scholar
  9. [9]
    D. J. Saunders, The Geometry of Jet Bundles, Cambridge University Press, 1989.Google Scholar
  10. [10]
    G. Terra and M. H. Kobayashi, On classical mechanical systems with non-linear constraints, Journal of Geometry and Physics, 49 (2004), pp. 385–417.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    —, On the variational mechanics with non-linear constraints, Journal de Mathématiques Pures et Appliquées, 83 (2004), pp. 629–671.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    T. Tsujishita, On variation bicomplexes associated to differential equations, Osaka J. Math., 19 (1982), pp. 311–363.MathSciNetMATHGoogle Scholar
  13. [13]
    W. M. Tulczyjew, The Euler-Lagrange resolution, vol. 836 of Lecture Notes in Mathematics, Springer-Verlag, New York, 1980, pp. 22–48.Google Scholar
  14. [14]
    A. M. Vinogradov, On the algebra-geometric foundation of Lagrangian field theory, Sov. Math. Dokl., 18 (1977), pp. 1200–1204.MATHGoogle Scholar
  15. [15]
    —, The c-spectral sequence, Lagrangian formalism and conservation laws i, ii, J. Math. Anal. Appl., 100 (1984), pp. 1–129.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2010

Authors and Affiliations

  1. 1.ISR and Departamento de MatemáticaInstituto Superior TécnicoLisbonPortugal
  2. 2.Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrazil

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