Noether theorem for actiondependent Lagrangian functions: conservation laws for nonconservative systems
Abstract
In the present work, we formulate a generalization of the Noether Theorem for actiondependent Lagrangian functions. The Noether's theorem is one of the most important theorems for physics. It is well known that all conservation laws, \textrm{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 nonconservative 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 actiondependent Lagrangian functions provides physically meaningful Lagrangian functions for a huge variety of nonconservative systems (classical and quantum). Consequently, the generalized Noether Theorem we present enable us to investigate conservation laws of nonconservative 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 arXiv:
 arXiv:1906.06182
 Bibcode:
 2019arXiv190606182L
 Keywords:

 Mathematical Physics;
 Mathematics  Optimization and Control;
 Physics  Classical Physics
 EPrint:
 accepted to publication on Nonlinear Dynamics. arXiv admin note: text overlap with arXiv:1803.08308