諾特第二定理

在數學和理論物理學中，諾特第二定理把作用量泛函的對稱性與微分方程系統聯繫起來。 物理系統的作用量S是所謂的拉格朗日函數L的積分，從作用量出發，可以通過最小作用量原理確定系統的行為。

具體地，該定理是說，如果一個作用量有由 k 個任意函數與它最高到m階的導數線性參數化的無窮小對稱性的無限維李代數，則L的泛函導數滿足一個包含k個方程的微分方程系統。

諾特第二定理可以用在規範理論中。規範理論是所有現代物理學場論的基本要素，例如通行的標準模型。

該定理以艾美·諾特的命名。

參見

• 諾特第一定理
• 諾特恆等式
• 規範對稱性（數學）

參考

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

延伸閲讀

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