Noether's second theorem


Noether's second theorem

In mathematics, 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.

Contents

See also

Notes

  1. ^ Noether E (1918), "Invariante Variationsprobleme", Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse 1918: 235–257, arXiv:physics/0503066, Bibcode 1971TTSP....1..186N, doi:10.1080/00411457108231446. 

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, 107 (2nd ed.), Springer-Verlag, ISBN 0-387-95000-1 

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