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Emmy Noether: Theorems
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"The well-known theorem of Emmy Noether plays a role of fundamental importance in many branches of theoretical physics. Because it provides a straightforward connection between the conservation laws of a physical theory and the invariances of the variational integral whose Euler-Lagrange equations are the [governing] equations of that theory, it may be said that Noether's theorem has placed the Lagrangian formulation in a position of primacy. In addition, the theorem brought about a situation whereby the search for conservation laws and selection rules has been reduced to the systematic study of the symmetries of a theory and the corresponding invariances of its Lagrangian."
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In 1918, Noether proved two theorems that formed a cornerstone for general relativity. These theorems validated certain relationships suspected by physicists of the time. One, now known as Noether's Theorem, established the equivalence between an invariance property and a conservation law. The other involved the relationship between an invariance and the existence of certain integrals of the equations of motion. The eminent German mathematician Hermann Weyl described Noether's contribution in the July 1935 Scripta Mathematica following her death: "For two of the most significant sides of the general theory of relativity theory she gave at that time the genuine and universal mathematical formulation."
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Noether's theorem is an amazing result which lets physicists get conserved quantities from symmetries of the laws of nature. Time translation symmetry gives conservation of energy; space translation symmetry gives conservation of momentum; rotation symmetry gives conservation of angular momentum, and so on.
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Noether's theorem is a central result in theoretical physics that shows that a conservation law can be derived from any differentiable symmetry. For example, the conservation of energy is a consequence of the fact that all laws of physics (including the values of the physical constants) are invariant under translation through time; they do not change as time passes.
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