Noether's theorem

Noether's theorem is a central result in theoretical physics that expresses the one-to-one correspondence between the symmetries and the conservation laws. This exact equivalence holds for all physical laws based upon the action principle defined over a symplectic space. It is named after the early 20th century mathematician Emmy Noether.

Related Topics:
Theoretical - Physics - Symmetries - Conservation law - Physical law - Action principle - Symplectic space - 20th century - Emmy Noether

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The word "symmetry" in the previous paragraph really means the covariance of the form that a physical law takes with respect to a one-dimensional Lie group of transformations which satisfies certain technical criteria. The conservation law of a physical quantity is usually expressed as a continuity equation.

Related Topics:
Covariance - Lie group - Conservation law - Physical quantity - Continuity equation

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The most important examples of the theorem are the following:

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• the energy is conserved if and only if the physical laws are invariant under time translations (if their form does not depend on time)
• the momentum is conserved iff the physical laws are invariant under spatial translations (if the laws do not depend on the position)
• the angular momentum is conserved iff the physical laws are invariant under rotations (if the laws do not care about the orientation); if only some rotations are allowed, only the corresponding components of the angular momentum vector are conserved

A Noether charge is a physical quantity conserved as an effect of a continuous symmetry of the underlying system.

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One theoretical use of the Noether charge is in calculating the entropy of stationary black holes{{fn|1}}.

Related Topics:
Entropy - Stationary black hole

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Noether's theorem: Mathematical statement of the theorem ►