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.
Mathematical formulation
First variation formula
Suppose that we have a dynamical system specified in terms of independent variables , dependent variables , and a Lagrangian function of some finite order . Here is the collection of all th order partial derivatives of the dependent variables. As a general rule, latin indices from the middle of the alphabet take the values , greek indices take the values , and the summation convention apply to them. Multiindex notation for the latin indices is also introduced as follows. A multiindex of length is an ordered list of ordinary indices. The length is denoted as . The summation convention does not directly apply to multiindices since the summation over lengths needs to be displayed explicitly, e.g.
Variational symmetries
A variation is an infinitesimal symmetry of the Lagrangian if under this variation. It is an infinitesimal quasi-symmetry if there is a current such that .
It should be remarked that it is possible to extend infinitesimal (quasi-)symmetries by including variations with as well, i.e. the independent variables are also varied. However such symmetries can always be rewritten so that they act only on the dependent variables. Therefore, in the sequel we restrict to so-called vertical variations where .
For Noether's second theorem, we consider those variational symmetries (called gauge symmetries) which are parametrized linearly by a set of arbitrary functions and their derivatives. These variations have the generic form
For these variations to be (exact, i.e. not quasi-) gauge symmetries of the Lagrangian, it is necessary that for all possible choices of the functions . If the variations are quasi-symmetries, it is then necessary that the current also depends linearly and differentially on the arbitrary functions, i.e. then , where
Noether's second theorem
The statement of Noether's second theorem is that whenever given a Lagrangian as above, which admits gauge symmetries parametrized linearly by arbitrary functions and their derivatives, then there exist linear differential relations between the Euler-Lagrange equations of .
Combining the first variation formula together with the fact that the variations are symmetries, we get
The expressions are differential in the Euler-Lagrange expressions, specifically we have
Converse result
A converse of the second Noether them can also be established. Specifically, suppose that the Euler-Lagrange expressions of the system are subject to differential relations
See also
Notes
- ^ 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): 186–207. arXiv:physics/0503066. Bibcode:1971TTSP....1..186N. doi:10.1080/00411457108231446. S2CID 119019843.
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.