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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

Abstract

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.

Keywords

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

Notes

Acknowledgements

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.

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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

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