Noether's Second Theorem is a result of mathematics and theoretical physics, proved by Emmy Noether alongside her more widely known first theorem, relating the symmetries of an action functional that depend on arbitrary functions of position, rather than on constant parameters, to a system of differential identities among the equations of motion. It governs the local, or gauge, symmetries found in field theories such as general relativity and electromagnetism, in contrast to the global symmetries and conserved quantities covered by Noether's first theorem.
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StatementWhenever a Lagrangian admits gauge symmetries parametrized linearly by q arbitrary functions and their derivatives, there exist q linear differential relations among the Euler-Lagrange equations of that Lagrangian. 1 Classification
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Source Noether's second theorem, Wikipedia
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Source Noether's second theorem, Wikipedia
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1. Noether's second theorem, Wikipedia
Noether's second theorem section
The statement of Noether's second theorem is that whenever given a Lagrangian L as above that admits gauge symmetries parametrized linearly by q arbitrary functions and their derivatives, then there exist q linear differential relations between the Euler-Lagrange equations of L.
References
Invariante Variationsprobleme, Nachr. D. Konig. Gesellsch. D. Wiss. Zu Gottingen, Math-phys. Klasse, 1918: 235-257.
In Branch: Mathematical Physics, Lead
Noether's second theorem relates symmetries of an action functional with a system of differential equations
Proved By: Emmy Noether, Lead
The theorem is named after its discoverer, Emmy Noether.
In Group: Classical Mechanics and Dynamics, Lead section
and theoretical physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations. The
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