Noether's second theorem

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In mathematics and theoretical physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations.[1] The action S of a physical system is an integral of a so-called Lagrangian function L, from which the system's behavior can be determined by the principle of least action.

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. Gauge theories are the basic elements of all modern field theories of physics, such as the prevailing Standard Model.

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  1. ^ Noether, Emmy (1918), "Invariante Variationsprobleme", Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse, 1918: 235–257
    Translated in Noether, Emmy (1971). "Invariant variation problems". Transport Theory and Statistical Physics. 1 (3): 186. arXiv:physics/0503066. Bibcode:1971TTSP....1..186N. doi:10.1080/00411457108231446.


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