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.

The theorem is named after its discoverer, Emmy Noether.

See also

• Noether's first theorem
• Noether identities
• Gauge symmetry (mathematics)

Notes

References

• Kosmann-Schwarzbach, Yvette (2010). The Noether theorems: Invariance and conservation laws in the twentieth century. Sources and Studies in the History of Mathematics and Physical Sciences. Springer-Verlag. ISBN 978-0-387-87867-6.
• Olver, Peter (1993). Applications of Lie groups to differential equations. Graduate Texts in Mathematics. Vol. 107 (2nd ed.). Springer-Verlag. ISBN 0-387-95000-1.
• Sardanashvily, G. (2016). Noether's Theorems. Applications in Mechanics and Field Theory. Springer-Verlag. ISBN 978-94-6239-171-0.

Further reading

• Noether, Emmy (1971). "Invariant Variation Problems". Transport Theory and Statistical Physics. 1 (3): 186–207. arXiv:physics/0503066. Bibcode:1971TTSP....1..186N. doi:10.1080/00411457108231446. S2CID 119019843.
• Fulp, Ron; Lada, Tom; Stasheff, Jim (2002). "Noether's variational theorem II and the BV formalism". arXiv:math/0204079.
• Bashkirov, D.; Giachetta, G.; Mangiarotti, L.; Sardanashvily, G (2008). "The KT-BRST Complex of a Degenerate Lagrangian System". Letters in Mathematical Physics. 83 (3): 237–252. arXiv:math-ph/0702097. Bibcode:2008LMaPh..83..237B. doi:10.1007/s11005-008-0226-y. S2CID 119716996.
• Montesinos, Merced; Gonzalez, Diego; Celada, Mariano; Diaz, Bogar (2017). "Reformulation of the symmetries of first-order general relativity". Classical and Quantum Gravity. 34 (20): 205002. arXiv:1704.04248. Bibcode:2017CQGra..34t5002M. doi:10.1088/1361-6382/aa89f3. S2CID 119268222.
• Montesinos, Merced; Gonzalez, Diego; Celada, Mariano (2018). "The gauge symmetries of first-order general relativity with matter fields". Classical and Quantum Gravity. 35 (20): 205005. arXiv:1809.10729. Bibcode:2018CQGra..35t5005M. doi:10.1088/1361-6382/aae10d. S2CID 53531742.