Symmetries and Noether's theorems
Abstract
Introduction Emmy Noether's greatest contributions to science were in algebra, but for physicists her name will always be remembered for her paper of 1918 on an invariance problem in the calculus of variations. The most celebrated part of this work, associated with her ‘first theorem’, has to do with the connection between continuous (global) symmetries in Lagrangian dynamics and conservation principles, though the main focus of the paper was the relationship between this and the second part of her paper, where she gives a systematic treatment of the more subtle and general case of continuous local symmetries (symmetries depending on arbitrary functions of the spacetime coordinates). The connection between global or ‘rigid’ symmetries and conservation principles in classical mechanics was hardly news in 1918. As Kastrup (1987) discusses in his historical review, it had been appreciated in the previous century by Lagrange, Hamilton, Jacobi, and Poincaré, and an anticipation of Noether's first theorem in the special cases of the 10-parameter Lorentz and Galilean groups had been given by Herglotz in 1911 and Engel in 1916, respectively. Noether's own contribution is often praised for its degree of generality, and not without reason. But interestingly it does not cover the cases in which the symmetry transformation preserves the Lagrangian or Lagrangian density only up to a divergence term. It does not therefore cover such cases as the boost symmetry in classical pre-relativistic…
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Symmetry transformations, such as local gauge symmetries, have empirical significance because they restrict the kinds of terms that can appear in the action and in the equations of motion.
“For example, Brading and Brown (2003: p. 99) have emphasised that symmetry transformations, such as local gauge symmetries, do have an empirical significance, because they restrict the kinds of terms that can appear in the action and in the equations of motion.”
From Noether's Theorems and Energy in General Relativity (2021) · cites this paper as Brading and Brown (2003: p. 99)
Recent discussions have concluded that the alleged lack of physical significance of the conservation laws derived from the second theorem, as claimed by Klein, is misleading.
“Recent discussions have also concluded that the alleged lack of physical significance of the conservation laws derived from the second theorem, as claimed by Klein (see Section 1.1), is misleading.”
From Noether's Theorems and Energy in General Relativity (2021) · cites this paper as Brading and Brown (2003: p. 99)
Noether's first theorem states that if the action is invariant with respect to a continuous group of transformations G_n generated by n constant parameters, then there are n linearly independent combinations of the field equations that are equal to the total divergence of a current.
“Noether's first theorem states that, if the action Eq. (25) is invariant with respect to a continuous group of transformations G n that is generated by n constant parameters ξ k (where k = 1 , . . . , n ), in the sense just indicated, then there are n linearly independent combinations of the field equations that are equal to the total divergence of a current, as follows:”
From Noether's Theorems and Energy in General Relativity (2021) · cites this paper as Brading and Brown (2003: pp. 91-97)
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Cited by 95
- Authors
- Brading, K., & Brown, H. R.
- Affiliations
- University of Oxford (GB)
- Venue
- Symmetries in Physics
- Publisher
- Cambridge University Press
- Published
- 2003-12-04
- Pages
- 89-109
- OpenAlex
- W91670585
- Cited by
- 95
- References
- 0
- Citations by year
- 2026: 2 · 2025: 3 · 2024: 2 · 2023: 21 · 2022: 8 · 2021: 1 · 2020: 1 · 2019: 2 · 2018: 3 · 2017: 5 · 2016: 6 · 2015: 7 · 2014: 6 · 2013: 3 · 2012: 1
- FWCI
- 15.59
- Type
- book chapter
- Language
- en
- Open access
- closed
- Locations
- Cambridge University Press eBooks (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and Classical Electrodynamics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Advanced Differential Geometry Research
- Keywords
- Noether's first theorem, Noether's second theorem, continuous symmetries, conservation laws, Lagrangian mechanics, calculus of variations, local symmetry, global symmetries, Emmy Noether, history of physics, boost symmetry
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