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Noether's theorem

Noether's theorem is a very crucial result in theoretical[?] physics that expresses the equivalence of two different properties of physical laws, the former being that of invariance of the form that the law takes with respect to any generalized transformation that the coordinate system (both spatial and temporal aspects taken into consideration) is subject to, and the latter being a conservation law of a physical quantity. It is named after the mathematician Emmy Noether of the early 20th century.

The formal statement of the theorem derives an expression for the physical quantity (and hence also defines it) that is conserved, from the condition of invariance alone. For example, the invariance of physical systems with respect to translation (when simply stated, it is just that the laws of physics don't vary with location in space) translates into the law of conservation of linear momentum. Invariance with respect to rotation gives law of conservation of angular momentum, invariance with respect to time gives the well known law of conservation of energy, et cetera. When it comes to quantum field theory, the invariance with respect to general gauge transformations gives the law of conservation of electric charge and so on. Thus, the result is a very important contribution to physics in general, as it helps to provide powerful insights into any general theory in physics, by just analyzing the various transformations that would make the form of the laws involved, invariant.

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