A geometric approach to the generalized Noether theorem. (12th February 2021)

• Record Type: Journal Article
• Authors: Bravetti, Alessandro; Garcia-Chung, Angel
• Is Part Of: Journal of physics. Volume 54:Number 9(2021)
• Publication Date: 2021-02-12
• Subjects: Noether theorem -- scaling symmetries -- contact Hamiltonian systems -- dissipative systems; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/abde78
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: We provide a geometric extension of the generalized Noether theorem for scaling symmetries recently presented by Zhang P-M et al (2020 Eur. Phys. J. Plus 135 223). Our version of the generalized Noether theorem has several positive features: it is constructed in the most natural extension of the phase space, allowing for the symmetries to be vector fields on such manifold and for the associated invariants to be first integrals of motion; it has a direct geometrical proof, paralleling the proof of the standard phase space version of Noether's theorem; it automatically yields an inverse Noether theorem; it applies also to a large class of dissipative systems; and finally, it allows for a much larger class of symmetries than just scaling transformations which form a Lie algebra, and are thus amenable to algebraic treatments.