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

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, \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 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 enable 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.

accepted to publication on Nonlinear Dynamics. arXiv admin note: text overlap with arXiv:1803.08308

References in corpus (4)

• Variational problems of Herglotz type with time delay: DuBois-Reymond condition and Noether's first theorem
• From an Action Principle for Action-dependent Lagrangians toward non-conservative Gravity: accelerating Universe without dark energy
• An Action Principle for Action-dependent Lagrangians: toward an Action Principle to non-conservative systems
• The action principle for dissipative systems

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• Practical Introduction to Action-Dependent Field Theories
• Fermat's Principle in General Relativity via Herglotz Variational Formalism