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Reference.org
Noether's second theorem
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Theoretical_physics/Cs2Y12rQ">theoretical physics</a>, Noether's second theorem relates symmetries of an <a href="/facts/Action_(physics)/NhWS2Mts">action</a> <a href="/facts/Functional_(mathematics)/3Cp3PHBR">functional</a> with a system of <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a>. The theorem is named after its discoverer, <a href="/facts/Emmy_Noether/agcYrVLt">Emmy Noether</a>. </p><p>The action <i>S</i> of a physical system is an <a href="/facts/Integral/V7lyV997">integral</a> of a so-called <a href="/facts/Lagrangian_mechanics/sTxyqWz8">Lagrangian</a> function <i>L</i>, from which the system's behavior can be determined by the <a href="/facts/Principle_of_least_action/QaBTpoKz">principle of least action</a>. Specifically, the theorem says that if the action has an infinite-dimensional <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> of infinitesimal symmetries parameterized linearly by <i>k</i> arbitrary functions and their derivatives up to order <i>m</i>, then the <a href="/facts/Functional_derivative/urNSIWYc">functional derivatives</a> of <i>L</i> satisfy a system of <i>k</i> differential equations. </p><p>Noether's second theorem is sometimes used in <a href="/facts/Gauge_theory/fSW3gggv">gauge theory</a>. Gauge theories are the basic elements of all modern <a href="/facts/Field_theory_(physics)/FJdAKeRA">field theories</a> of physics, such as the prevailing <a href="/facts/Standard_Model/LhVGf9ow">Standard Model</a>. </p>