Nonlinear Dynamics

, Volume 83, Issue 1–2, pp 457–461 | Cite as

Geometric approach to dynamics obtained by deformation of Lagrangians

Original Paper


The relationship of equations of motion of a Lagrangian \(\phi (L)\) to those of L is studied, and the question of the existence of a function \(\phi \) such that \(\phi (L)\) is dynamically equivalent to L is answered.


Non-standard Lagrangians Inverse problem Deformation of Lagrangians Modified Euler–Lagrange equations 

Mathematics Subject Classification

53Z05 70G45 70H03 


  1. 1.
    Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Benjamin, Reading MA (1978)MATHGoogle Scholar
  2. 2.
    Crampin, M.: On the differential geometry of the Euler–Lagrange equations and the inverse problem in Lagrangian dynamics. J. Phys. A Math. Gen. 14, 2567–2575 (1981)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Crampin, M.: Tangent bundle geometry for Lagrangian dynamics. J. Phys. A Math. Gen. 16, 3755–3772 (1983)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Crampin, M., Pirani, F.A.E.: Applicable Differential Geometry. University Press, Cambridge (1986)MATHGoogle Scholar
  5. 5.
    Cariñena, J.F., Ibort, A., Marmo, G., Morandi, G.: Geometry from Dynamics, Classical and Quantum. Springer, Berlin (2015). ISBN 978-94-017-9219-6CrossRefMATHGoogle Scholar
  6. 6.
    Biedenharn, L.C.: The quantum group \(SU_q(2) \) and a \(q\)-analogue of the boson operators. J. Phys. A Math. Gen. 22, L873–L878 (1989)Google Scholar
  7. 7.
    Macfarlane, A.J.: On \(q\)-analogues of the quantum harmonic oscillator and the quantum group \(SU(2)_q\). J. Phys. A Math. Gen. 22, 4581–4588 (1989)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Man’ko, V.I., Marmo, G., Sudarshan, E.C.G., Zaccaria, F.: \(f\)-oscillators and nonlinear coherent states. Phys. Scr. 55, 528–541 (1997)CrossRefGoogle Scholar
  9. 9.
    D’Avanzo, A., Marmo, G.: Reduction and unfolding: the Kepler problem. Int. J. Geom. Methods Mod. Phys. 2, 83–109 (2005)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    D’Avanzo, A., Marmo, G., Valentino, A.: Reduction and unfolding for quantum systems: the hydrogen atom. Int. J. Geom. Methods Mod. Phys. 2, 1043–1062 (2005)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Marle, C.M.: A property of conformally Hamiltonian vector fields: application to the Kepler problem. J. Geom. Mech. 4, 181–206 (2012)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Helmholtz, H.: Über die physikalische bedeutung des princips der kleinsten wirking. J. Reine Angew. Math. 100, 137–166 (1887)MathSciNetMATHGoogle Scholar
  13. 13.
    Douglas, J.: Solution of the inverse problem of the calculus of variations. Trans. Am. Math. Soc. 50, 71–128 (1941)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Currie, D.G., Saletan, E.J.: \(q\)-equivalent particle Hamiltonians. The classical one-dimensional case. J. Math. Phys. 7, 967–974 (1966)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Hojman, S., Harleston, H.: Equivalent Lagrangians: multidimensional case. J. Math. Phys. 22, 1414–1419 (1981)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Cariñena, J.F., Ibort, L.A.: Non-Noether constants of motion. J. Phys. A Math. Gen. 16, 1–7 (1983)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Cariñena, J.F., Rañada, M.F., Santander, M.: Lagrangian formalism for nonlinear second-order Riccati systems: one-dimensional integrability and two-dimensional superintegrability. J. Math. Phys. 46, 062703 (2005)Google Scholar
  18. 18.
    Cariñena, J.F., Guha P, P., Rañada, M.F.: Higher-order Abel equations: Lagrangian formalism, first integrals and Darboux polynomials. Nonlinearity 22, 2953–2969 (2009)Google Scholar
  19. 19.
    Musielak, Z.E.: Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. J. Phys. A Math. Theor. 41, 055205 (2008)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Musielak, Z.E., Roy, D., Swift, L.D.: Method to derive Lagrangian and Hamiltonian for a nonlinear dynamical system with variable coefficients. Chaos Solitons Fractals 38, 894–902 (2008)CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Musielak, Z.E.: General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems. Chaos Solitons Fractals 42, 2645–2652 (2009)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Cieśliński, J.L., Nikiciuk, T.: A direct approach to the construction of standard and non-standard Lagrangians for dissipative-like dynamical systems with variable coefficients. J. Phys. A Math. Theor. 43, 175205 (2010)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Saha, A., Talukdar, B.: On the non-standard Lagrangian equations. arXiv: 1301.2667
  24. 24.
    Saha, A., Talukdar, B.: Inverse variational problem for non-standard Lagrangians. Rep. Math. Phys. 73, 299–309 (2014)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    El-Nabulsi, R.A.: Non-standard Lagrangians in rotational dynamics and the modified Navier–Stokes equation. Nonlinear Dyn. 79, 2055–2068 (2015)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    El-Nabulsi, R.A.: Non-linear dynamics with non-standard Lagrangians. Qual. Theory Dyn. Syst. 13, 273–291 (2013)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    El-Nabulsi, R.A.: Modified Proca equation and modified dispersion relation from a power-law Lagrangian functional. Indian J. Phys. 87, 465–470 (2013)CrossRefGoogle Scholar
  28. 28.
    El-Nabulsi, R.A.: Electrodynamics of relativistic particles through non-standard Lagrangian. J. At. Mol. Sci. 5, 268–278 (2014)MathSciNetGoogle Scholar
  29. 29.
    El-Nabulsi, R.A.: A generalized nonlinear oscillator from non-standard degenerate Lagrangians and its consequent Hamiltonian Formalism. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 84, 563–569 (2014)CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    El-Nabulsi, R.A.: Non-standard power-law Lagrangians in classical and quantum dynamics. Appl. Math. Lett. 43, 120–127 (2015)CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Cariñena, J.F., Gheorghiu, I., Martínez, E., Santos, P.: Conformal Killing vector fields and a virial theorem. J. Phys. A Math. Theor. 47, 465206 (18pp) (2014)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Saunders, D.J.: Homogeneous Lagrangian systems. Rep. Math. Phys. 51, 315–324 (2003)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Departamento de Física Teórica, Facultad de CienciasUniversidad de ZaragozaZaragozaSpain
  2. 2.Departamento de Física, Facultad de CienciasUniversidad de OviedoOviedoSpain

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