Noether theorem for action-dependent Lagrangian functions: conservation laws for non-conservative systems

  • M. J. LazoEmail author
  • J. Paiva
  • G. S. F. Frederico
Original Paper


In the present work, we formulate a generalization of the Noether Theorem for action-dependent Lagrangian functions. The Noether’s theorem is one of the most important theorems for physics. It is well known that all conservation laws, e.g., conservation of energy and momentum, are directly related to the invariance of the action under a family of transformations. However, the classical Noether theorem cannot be applied to study non-conservative systems because it is not possible to formulate physically meaningful Lagrangian functions for this kind of systems in the classical calculus of variation. On the other hand, recently it was shown that an Action Principle with action-dependent Lagrangian functions provides physically meaningful Lagrangian functions for a huge variety of non-conservative systems (classical and quantum). Consequently, the generalized Noether Theorem we present enables us to investigate conservation laws of non-conservative systems. In order to illustrate the potential of application, we consider three examples of dissipative systems and we analyze the conservation laws related to spacetime transformations and internal symmetries.


Action-dependent Lagrangian functions Noether theorem Non-conservative systems Herglotz variational problem 



This work was partially supported by CNPq and CAPES (Brazilian research funding agencies).

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Bauer, P.S.: Dissipative dynamical systems I. Proc. Natl. Acad. Sci 17, 311 (1931). CrossRefzbMATHGoogle Scholar
  2. 2.
    Stevens, K.W.H.: The wave mechanical damped harmonic oscillator. Proc. Phys. Soc. 72, 1027 (1958). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Havas, P.: The range of application of the Lagrange formalism—I. Nuovo Cimento Suppl. 5, 363 (1957). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Negro, F., Tartaglia, A.: The quantization of quadratic friction. Phys. Lett. A 77, 1 (1980). CrossRefGoogle Scholar
  5. 5.
    Negro, F., Tartaglia, A.: Quantization of motion in a velocity-dependent field: the \(v^2\) case. Phys. Rev. A 23, 1591 (1981). MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brinati, J.R., Mizrahi, S.S.: Quantum friction in the c-number picture: the damped harmonic oscillator. J. Math. Phys. 21, 2154 (1980). MathSciNetCrossRefGoogle Scholar
  7. 7.
    Tartaglia, A.: Non-conservative forces, Lagrangians and quantisation. Eur. J. Phys. 4, 231 (1983). CrossRefGoogle Scholar
  8. 8.
    Bateman, H.: On dissipative systems and related variational principles. Phys. Rev. 38, 815 (1931). CrossRefzbMATHGoogle Scholar
  9. 9.
    Morse, P.M., Feshbach, H.: Methods of Theoretical Physics. McGraw-Hill, New York (1953)zbMATHGoogle Scholar
  10. 10.
    Feshbach, H., Tikochinsky, Y.: Quantization of the damped harmonic oscillator. Trans. N. Y. Acad. Sci. 38, 44 (1977). CrossRefGoogle Scholar
  11. 11.
    Celeghini, E., Rasetti, M., Tarlini, M., Vitiello, G.: SU(1,1) squeezed states as damped oscillators. Mod. Phys. Lett. B 3, 1213 (1989). MathSciNetCrossRefGoogle Scholar
  12. 12.
    Celeghini, E., Rasetti, H., Vitiello, G.: Quantum dissipation. Ann. Phys. (N.Y.) 215, 156 (1992). MathSciNetCrossRefGoogle Scholar
  13. 13.
    Vujanovic, B.D., Jones, S.E.: Variational Methods in Nonconservative Phenomena. Academic Press, San Diego (1989)zbMATHGoogle Scholar
  14. 14.
    Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890 (1996). MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lazo, M.J., Krumreich, C.E.: The action principle for dissipative systems. J. Math. Phys. 55, 122902 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lazo, M.J., Paiva, J., Amaral, J.T.S., Frederico, G.S.F.: Action principle for action-dependent Lagrangians toward nonconservative gravity: accelerating universe without dark energy. Phys. Rev. D 95, 101501(R) (2017). CrossRefGoogle Scholar
  17. 17.
    Lazo, M.J., Paiva, J., Amaral, J.T.S., Frederico, G.S.F.: An action principle for action-dependent Lagrangians: toward an action principle to non-conservative systems. J. Math. Phys. 59, 032902 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Herglotz, G.: Berührungstransformationen. Lectures at the University of Göttingen, Göttingen (1930)Google Scholar
  19. 19.
    Guenther, R.B., Guenther, C.M., Gottsch, J.A.: The Herglotz Lectures on Contact Transformations and Hamiltonian Systems. Lecture Notes in Nonlinear Analysis, vol. 1. Juliusz Schauder Center for Nonlinear Studies, Nicholas Copernicus University, Torún (1996)zbMATHGoogle Scholar
  20. 20.
    Georgieva, B., Guenther, R., Bodurov, T.: Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem. J. Math. Phys. 44, 3911 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Georgieva, B., Guenther, R.: First Noether-type theorem for the generalized variational principle of Herglotz. Topol. Methods Nonlinear Anal. 20(1), 261–273 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Santos, S.P.S., Martins, N., Torres, D.F.M.: Variational problems of Herglotz type with time delay: DuBois–Reymond condition and Noether’s first theorem. Discrete Contin. Dyn. Syst. A 35(9), 4593 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhang, Y.: Variational problem of Herglotz type for Birkhoff system and its Noether’s theorems. Acta Mech. 228(4), 1–12 (2017). MathSciNetGoogle Scholar
  24. 24.
    Zhang, Y.: Noether’s theorem for a time-delayed Birkhoffian system of Herglotz type. Int. J. Nonlinear Mech. 101, 36–43 (2018). CrossRefGoogle Scholar
  25. 25.
    Tian, X., Zhang, Y.: Noether’s theorem and its inverse of Birkhoffian system in event space based on Herglotz variational problem. Int. J. Theor. Phys. 57(3), 887–897 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tian, X., Zhang, Y.: Noether symmetry and conserved quantities of fractional Birkhoffian system in terms of Herglotz variational problem. Commun. Theor. Phys. 70(3), 280–288 (2018). CrossRefGoogle Scholar
  27. 27.
    Tian, X., Zhang, Y.: Noether’s theorem for fractional Herglotz variational principle in phase space. Chaos Solitons Fractals 119, 50–54 (2019). MathSciNetCrossRefGoogle Scholar
  28. 28.
    Symon, K.R.: Mechanics, 3rd edn. Addison-Wesley Publishing Company Inc., Reading, MA (1971)zbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Instituto de Matemática, Estatística e FísicaUniversidade Federal do Rio GrandeRio GrandeBrazil
  2. 2.Universidade Federal do CearáRussasBrazil

Personalised recommendations