# Noether's second theorem > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Noether's_second_theorem > Markdown URL: https://mediated.wiki/source/Noether's_second_theorem.md > Source: https://en.wikipedia.org/wiki/Noether's_second_theorem > Source revision: 1356504919 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) In [mathematics](/source/Mathematics) and [theoretical physics](/source/Theoretical_physics), **Noether's second theorem** relates symmetries of an [action](/source/Action_(physics)) [functional](/source/Functional_(mathematics)) with a system of [differential equations](/source/Differential_equation).[1] The theorem is named after its discoverer, [Emmy Noether](/source/Emmy_Noether). The action *S* of a [physical system](/source/Physical_system) is an [integral](/source/Integral) of a so-called [Lagrangian](/source/Lagrangian_mechanics) function *L*, from which the system's behavior can be determined by the [principle of least action](/source/Principle_of_least_action). Specifically, the theorem says that if the action has an infinite-dimensional [Lie algebra](/source/Lie_algebra) of [infinitesimal](/source/Infinitesimal) symmetries parameterized linearly by *k* arbitrary functions and their derivatives up to order *m*, then the [functional derivatives](/source/Functional_derivative) of *L* satisfy a system of *k* differential equations. Noether's second theorem is sometimes used in [gauge theory](/source/Gauge_theory). Gauge theories are the basic elements of all modern [field theories](/source/Field_theory_(physics)) of physics, such as the prevailing [Standard Model](/source/Standard_Model). ## Mathematical formulation ### First variation formula Suppose that we have a [dynamical system](/source/Dynamical_system) specified in terms of m independent variables x=(x^1,\dots,x^m ), n dependent variables u=(u^1,\dots, u^n ), and a [Lagrangian](/source/Lagrangian_(field_theory)) function L(x,u,u_{(1)}\dots,u_{(r)}) of some finite order r. Here u_{(k)}=(u^\sigma_{i_1...i_k})=(d_{i_1}\dots d_{i_k}u^\sigma) is the collection of all kth order partial derivatives of the dependent variables. As a general rule, Latin indices i,j,k,\dots from the middle of the alphabet take the values 1,\dots,m, Greek indices take the values 1,\dots,n, and the [summation convention](/source/Einstein_notation) apply to them. Multiindex notation for the Latin indices is also introduced as follows. A *multiindex* I of length k is an ordered list I=(i_1,\dots,i_k ) of k ordinary indices. The length is denoted as \left|I \right|=k. The summation convention does not directly apply to multiindices since the summation over lengths needs to be displayed explicitly, e.g. \sum_{|I|=0}^r f_I g^I = fg + f_i g^i + f_{ij} g^{ij} + \dots + f_{i_1...i_r} g^{i_1...i_r}. The variation of the Lagrangian with respect to an arbitrary variation \delta u^\sigma of the dependent variables is \delta L = \frac{\partial L}{\partial u^\sigma} \delta u^\sigma + \frac{\partial L}{\partial u^\sigma_i}\delta u ^\sigma_i + \dots + \frac{\partial L}{\partial u^\sigma_{i_1...i_r}} \delta u^\sigma_{i_1...i_r} = \sum_{|I|=0}^r \frac{\partial L}{\partial u^\sigma_I}\delta u^\sigma_I, and applying the [inverse product rule of differentiation](/source/Integration_by_parts) we get \delta L = E_\sigma \delta u^\sigma + d_i\left(\sum_{|I|=0}^{r-1} P^{iI}_\sigma \delta u^\sigma_I\right), where E_\sigma=\frac{\partial L}{\partial u^\sigma} - d_i \frac{\partial L}{\partial u^\sigma_i} +\dots + (-1)^r d_{i_1}\dots d_{i_r} \frac{\partial L}{\partial u^\sigma_{i_1...i_r}} = \sum_{|I|=0}^r (-1)^{|I|}d_I \frac{\partial L}{\partial u^\sigma_I} are the [Euler-Lagrange expressions](/source/Euler%E2%80%93Lagrange_equation) of the Lagrangian, and the coefficients P^I_\sigma (Lagrangian momenta) are given by P^I_\sigma = \sum_{|J|=0}^{r-|I|}(-1)^{|J|}d_J\frac{\partial L}{\partial u^\sigma_{IJ}}. ### Variational symmetries A variation \delta u^\sigma = X^\sigma(x,u,u_{(1)},\dots) is an *infinitesimal symmetry* of the Lagrangian L if \delta L = 0 under this variation. It is an infinitesimal *quasi-symmetry* if there is a current K^i=K^i(x,u,\dots) such that \delta L = d_i K^i. It should be remarked that it is possible to extend infinitesimal (quasi-)symmetries by including variations with \delta x^i \neq 0 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 \delta x^i = 0. For Noether's second theorem, we consider those variational symmetries (called *[gauge symmetries](/source/Gauge_symmetry_(mathematics))*) which are parametrized linearly by a set of arbitrary functions and their derivatives. These variations have the generic form \delta_\lambda u^\sigma = R^\sigma_a \lambda^a + R^{\sigma,i}_{a}\lambda^a_{i} + \dots + R^{\sigma,i_1...i_s}_a \lambda^a_{i_1...i_s} = \sum_{|I|=0}^s R^{\sigma,I}_a \lambda^a_I, where the coefficients R^{\sigma,I}_a can depend on the independent and dependent variables as well as the derivatives of the latter up to some finite order, the \lambda^a = \lambda^a (x) are arbitrarily specifiable functions of the independent variables, and the Latin indices a,b,\dots take the values 1,\dots,q, where q is some positive integer. For these variations to be (exact, i.e. not quasi-) gauge symmetries of the Lagrangian, it is necessary that \delta_\lambda L = 0 for all possible choices of the functions \lambda^a (x). 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 \delta_\lambda L =d_i K^i_\lambda, where K^i_\lambda = K^i_a \lambda^a + K^{i,j}_a \lambda^a_{j} + K^{i,j_1j_2}_a \lambda^a_{j_1j_2}\dots For simplicity, we will assume that all gauge symmetries are exact symmetries, but the general case is handled similarly. ### Noether's second theorem The statement of Noether's second theorem is that whenever given a Lagrangian L as above that admits gauge symmetries \delta_\lambda u^\sigma parametrized linearly by q arbitrary functions and their derivatives, then there exist q linear differential relations between the Euler-Lagrange equations of L. Combining the first variation formula together with the fact that the variations \delta_\lambda u^\sigma are symmetries, we get 0 = E_\sigma \delta_\lambda u^\sigma + d_i W^i_\lambda,\quad W^i_\lambda = \sum_{|I|=0}^r P^{iI}_\sigma \delta_\lambda u^\sigma, where on the first term proportional to the Euler-Lagrange expressions, further integrations by parts can be performed as E_\sigma \delta_\lambda u^\sigma = \sum_{|I|=0}^s E_\sigma R^{\sigma, I}_a \lambda^a_I = Q_a\lambda^a + d_i\left(\sum_{|I|=0}^{s-1} Q^{iI}_a \lambda^a_I \right), where Q^I_a = \sum_{|J|=0}^{s-|I|}(-1)^{|J|}d_J\left( E_\sigma R^{\sigma,IJ}_a \right), in particular for |I| = 0, Q_a = E_\sigma R^\sigma_a - d_i\left( E_\sigma R^{\sigma,i}_a\right)+\dots + (-1)^s d_{i_1}\dots d_{i_s}\left( E_\sigma R^{\sigma,i_1...i_s}_a \right) = \sum_{|I|=0}^s (-1)^{|I|} d_I\left(E_\sigma R^{\sigma,I}_a\right) . Hence, we have an *[off-shell](/source/On_shell_and_off_shell)* relation 0 = Q_a\lambda^a + d_i S^i_\lambda, where S^i_\lambda = H^i_\lambda + W^i_\lambda, with H^i_\lambda = \sum_{|I|=0}^{s-1} Q^{iI}_a\lambda^a_I. This relation is valid for any choice of the gauge parameters \lambda^a (x). Choosing them to be compactly supported, and integrating the relation over the manifold of independent variables, the integral total divergence terms vanishes due to [Stokes' theorem](/source/Generalized_Stokes_theorem). Then from the [fundamental lemma of the calculus of variations](/source/Fundamental_lemma_of_the_calculus_of_variations), we obtain that Q_a\equiv 0 identically as *off-shell* relations (in fact, since the Q_a are linear in the Euler-Lagrange expressions, they necessarily vanish on-shell). Inserting this back into the initial equation, we also obtain the off-shell conservation law d_i S^i_\lambda = 0. The expressions Q_a are differential in the Euler-Lagrange expressions, specifically we have Q_a = \mathcal D_a[E] = \sum_{|I|=0}^s (-1)^{|I|} d_I\left(E_\sigma R^{\sigma,I}_a\right) = \sum_{|I|=0}^s F^{\sigma, I}_a d_I E_\sigma, where F^{\sigma,I}_a = \sum_{|J|=0}^{s-|I|} \binom{|I|+|J|}{|I|} (-1)^{|I|+|J|} d_J R^{\sigma,IJ}_a. Hence, the equations 0 = \mathcal{D}_a[E] are q differential relations to which the Euler-Lagrange expressions are subject to, and therefore the Euler-Lagrange equations of the system are not independent. ### Converse result A converse of the second Noether theorem can also be established. Specifically, suppose that the Euler-Lagrange expressions E_\sigma of the system are subject to q differential relations 0 = \mathcal D_a[E] = \sum_{|I|=0}^s F^{\sigma,I}_a d_I E_\sigma. Letting \lambda = (\lambda^1,\dots,\lambda^q ) be an arbitrary q-[tuple](/source/Tuple) of functions, the [formal adjoint](/source/Differential_operator#Formal_adjoint_in_one_variable) of the operator \mathcal{D}_a acts on these functions through the formula E_\sigma (\mathcal{D}^+)^\sigma[\lambda] - \lambda^a\mathcal{D}_a [E] = d_i B^i_\lambda, which defines the adjoint operator (\mathcal{D}^+)^\sigma uniquely. The coefficients of the adjoint operator are obtained through integration by parts as before, specifically (\mathcal{D}^+)^\sigma [\lambda] = \sum_{|I|=0}^s R^{\sigma,I}_a \lambda^a_I, where R^{\sigma,I}_a = \sum_{|J|=0}^{s-|I|} (-1)^{|I|+|J|} \binom{|I|+|J|}{|I|} d_J F^{\sigma,IJ}_a. Then the definition of the adjoint operator together with the relations 0 = \mathcal{D}_a [E] state that for each q-tuple of functions \lambda, the value of the adjoint on the functions when contracted with the Euler-Lagrange expressions is a total divergence, viz. E_\sigma(\mathcal{D}^+)^\sigma [\lambda] = d_i B^i_\lambda, therefore if we define the variations \delta_\lambda u^\sigma := (\mathcal{D}^+)^\sigma[\lambda]=\sum_{|I|=0}^s R^{\sigma,I}_a\lambda^a_I, the variation \delta_\lambda L = E_\sigma \delta_\lambda u^\sigma + d_i W^i_\lambda = d_i\left(B^i_\lambda + W^i_\lambda\right) of the Lagrangian is a total divergence, hence the variations \delta_\lambda u^\sigma are quasi-symmetries for every value of the functions \lambda^a. ## See also - [Noether's first theorem](/source/Noether's_first_theorem) - [Noether identities](/source/Noether_identities) - [Gauge symmetry (mathematics)](/source/Gauge_symmetry_(mathematics)) ## Notes 1. Noether, Emmy (1918), ["Invariante Variationsprobleme"](https://de.wikisource.org/wiki/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](/source/Transport_Theory_and_Statistical_Physics)*. **1** (3): 186–207. [arXiv:physics/0503066](https://arxiv.org/abs/physics/0503066). [Bibcode:1971TTSP....1..186N](https://ui.adsabs.harvard.edu/abs/1971TTSP....1..186N). [doi:10.1080/00411457108231446](https://doi.org/10.1080/00411457108231446). [S2CID 119019843](https://api.semanticscholar.org/CorpusID: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](/source/Springer_Science%2BBusiness_Media). ISBN 978-0-387-87867-6. - Olver, Peter (1993). *Applications of Lie groups to differential equations*. Vol. 107. [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics). 2nd ed. [Springer-Verlag](/source/Springer_Science%2BBusiness_Media). ISBN 0-387-95000-1. - Sardanashvily, G. (2016). *Noether's Theorems. Applications in Mechanics and Field Theory*. [Springer-Verlag](/source/Springer_Science%2BBusiness_Media). ISBN 978-94-6239-171-0. ## Further reading - Noether, Emmy (1971). "Invariant Variation Problems". *[Transport Theory and Statistical Physics](/source/Transport_Theory_and_Statistical_Physics)*. **1** (3): 186–207. [arXiv:physics/0503066](https://arxiv.org/abs/physics/0503066). [Bibcode:1971TTSP....1..186N](https://ui.adsabs.harvard.edu/abs/1971TTSP....1..186N). [doi:10.1080/00411457108231446](https://doi.org/10.1080/00411457108231446). [S2CID 119019843](https://api.semanticscholar.org/CorpusID:119019843) - Fulp, Ron; Lada, Tom; Stasheff, Jim (2002). "Noether's variational theorem II and the BV formalism". [arXiv:math/0204079](https://arxiv.org/abs/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](https://arxiv.org/abs/math-ph/0702097). [Bibcode:2008LMaPh..83..237B](https://ui.adsabs.harvard.edu/abs/2008LMaPh..83..237B). [doi:10.1007/s11005-008-0226-y](https://doi.org/10.1007/s11005-008-0226-y). [S2CID 119716996](https://api.semanticscholar.org/CorpusID: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](https://arxiv.org/abs/1704.04248). [Bibcode:2017CQGra..34t5002M](https://ui.adsabs.harvard.edu/abs/2017CQGra..34t5002M). [doi:10.1088/1361-6382/aa89f3](https://doi.org/10.1088/1361-6382/aa89f3). [S2CID 119268222](https://api.semanticscholar.org/CorpusID: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](https://arxiv.org/abs/1809.10729). [Bibcode:2018CQGra..35t5005M](https://ui.adsabs.harvard.edu/abs/2018CQGra..35t5005M). [doi:10.1088/1361-6382/aae10d](https://doi.org/10.1088/1361-6382/aae10d). [S2CID 53531742](https://api.semanticscholar.org/CorpusID:53531742) --- Adapted from the Wikipedia article [Noether's second theorem](https://en.wikipedia.org/wiki/Noether's_second_theorem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Noether's_second_theorem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.