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Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem ) published by mathematician Emmy Noether in 1918. [ 1 ]
Specifically, the theorem says that if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by k arbitrary functions and their derivatives up to order m, then the functional derivatives of L satisfy a system of k differential equations. Noether's second theorem is sometimes used in gauge theory.
This is an accepted version of this page This is the latest accepted revision, reviewed on 31 December 2024. Law of physics and chemistry This article is about the law of conservation of energy in physics. For sustainable energy resources, see Energy conservation. Part of a series on Continuum mechanics J = − D d φ d x {\displaystyle J=-D{\frac {d\varphi }{dx}}} Fick's laws of diffusion ...
This is an instance of Noether's theorem. Here, the conserved quantity is the stress–energy tensor , which is only conserved on shell, that is, if the equations of motion are satisfied. References
Noether's theorem: Every continuous symmetry in a physical system has a corresponding conservation law. Occam's razor: explanations should never multiply causes without necessity. ("Entia non sunt multiplicanda praeter necessitatem.") When two or more explanations are offered for a phenomenon, the simplest full explanation is preferable.
Generally, the correspondence between continuous symmetries and conservation laws is given by Noether's theorem. The form of the fundamental quantum operators, for example the energy operator as a partial time derivative and momentum operator as a spatial gradient , becomes clear when one considers the initial state, then changes one parameter ...
Noether's work in abstract algebra and topology was influential in mathematics, while Noether's theorem has widespread consequences for theoretical physics and dynamical systems. Noether showed an acute propensity for abstract thought, which allowed her to approach problems of mathematics in fresh and original ways. [ 40 ]
No wandering domain theorem (ergodic theory) Noether's theorem (Lie groups, calculus of variations, differential invariants, physics) Noether's second theorem (calculus of variations, physics) Noether's theorem on rationality for surfaces (algebraic surfaces) Non-squeezing theorem (symplectic geometry) Norton's theorem (electrical networks)