Noether theorem

Noether's first theorem establishes a connection between the infinitesimal symmetries of a functional of the form

$$ A ( u ( x) ) = \int\limits L ( x , u ( x) , u _ {,j} ( x) ) d ^ {n} x , $$

where $ x = ( x ^ {1} \dots x ^ {n} ) $ are independent variables, $ u ( x) = ( u ^ {1} ( x) \dots u ^ {N} ( x) ) $ are functions defined in a certain domain $ D \subset \mathbf R ^ {n} $, $ u _ {,j} =(\partial / \partial x ^ {j} ) ( u ( x) ) $ are their partial derivatives, and $ L $ is a certain function (the Lagrangian), and the conservation laws for the corresponding system of Euler–Lagrange equations

$$ \frac{\delta L }{\delta u ^ {a} } \equiv \ \frac{\partial L }{\partial u ^ {a} } - \frac{d }{d x ^ {i} } \frac{\partial L }{\partial u _ {,i} ^ {a} } = 0 , $$

which gives necessary conditions for an extremum of $ A $. Namely, to an infinitesimal symmetry $ Z $, that is, a vector field

$$ Z = X ^ {i} ( x) \frac \partial {\partial x ^ {i} } + U ^ {a} ( x , u ) \frac \partial {\partial u ^ {a} } $$

that generates a one-parameter group of transformations preserving $ A $, corresponds the conservation law

$$ \nu _ {Z} = \ \left [ L X ^ {i} + ( U ^ {a} - u _ {,j} ^ {a} X ^ {j} ) \frac{\partial L }{\partial u _ {,i} ^ {a} } \right ] d x ^ {1} \wedge \dots \wedge \widehat{d} x ^ {i} \wedge \dots \wedge d x ^ {n} $$

(where the symbol $ \widehat{ {}} $ indicates the omission of the corresponding factor), that is, an $ ( n - 1 ) $- form depending on $ u ( x) $ that is closed when $ u ( x) $ satisfies the Euler–Lagrange equations.

In field theory, where $ n = 4 $ and the coordinates $ x $ are interpreted as space-time coordinates, $ A $ is called the action and $ u ( x) $ the field. To fields $ u ( x) $ providing an extremum of the action functional correspond physically realizable fields with a given Lagrange function. If such a field $ u ( x) $ vanishes on the boundary of $ D $, then by Stokes' theorem the integral of the conservation law $ v $ over a hypersurface $ D \cap \{ x ^ {1} = c \} $ does not depend on the choice of $ c $. In particular, if $ x ^ {1} $ is the time coordinate, then this integral yields a quantity that is preserved in the course of time (whence the name conservation law).

The invariance of the Lagrange function of distinct physical fields under parallel translations and Lorentz transformations (which is a consequence of the homogeneity and isotropy of Minkowski space-time) leads, by Noether's theorem, to the energy-momentum tensor and the angular momentum tensor of the field and to corresponding conservations laws for the energy, momentum and angular momentum of the motion. Invariance of the action functional of the electromagnetic field under gauge transformations leads to the conservation law for electric charge. Similarly, invariance of the Lagrangian of some field under gauge transformations yields conservation laws for various charges.

In classical mechanics, $ n = 1 $ and the coordinate $ x ^ {1} $ is interpreted as time. If the Lagrange function does not depend explicitly on $ x ^ {1} $, then the vector field $ \partial / \partial x ^ {1} $ is a symmetry, and Noether's theorem leads to the law of conservation of energy. For a mechanical system whose motion can be described as geodesic motion in some Riemannian metric, the symmetries of the corresponding action functional are Killing vector (or, more generally, Killing tensor) fields. In this case the conservation law furnished by Noether's theorem means geometrically that the magnitude of the projection of the Killing vector field in the direction of a geodesic is constant along it. The general modern formulation of Noether's theorem in the language of fibre bundles consists in the following. Let $ \pi : E \rightarrow M $ be a vector bundle over an $ n $- dimensional manifold $ M $ with a fixed volume $ n $- form $ \omega \in \Lambda ^ {n} ( M) $, and let $ \pi _ {k} : J ^ {k} E \rightarrow M $ be the vector bundle of $ k $- jets of sections of $ \pi $. If $ x ^ {i} $ are local coordinates in $ M $ in which $ \omega $ becomes $ \omega = d x ^ {1} \wedge \dots \wedge d x ^ {n} $, and if $ x ^ {i} , u ^ {a} $ are local coordinates in $ E $, then in $ J ^ {k} E $ one has local coordinates $ x ^ {i} , u ^ {a} , u ^ \alpha $, where $ \alpha = ( \alpha _ {1} \dots \alpha _ {n} ) $ is a multi-index and $ | \alpha | = \alpha _ {1} + \dots + \alpha _ {n} \leq k $. The value of the coordinate $ u ^ \alpha $ on the $ k $- jet $ J _ {x _ {0} } ^ {k} u ( x) $ of the section $ u ( x) $ of $ \pi $ is

$$ u _ {, \alpha } ^ {a} ( x _ {0} ) = \ \left ( \frac \partial {\partial x _ {1} } \right ) ^ {\alpha _ {1} } \dots \left ( \frac \partial {\partial x ^ {n} } \right ) ^ {\alpha _ {n} } u ^ {a} ( x _ {0} ) . $$

A smooth function $ L : J ^ {k} E \rightarrow \mathbf R $ determines an action functional $ A $ that associates with a section $ s : x \rightarrow u ( x) $ the number

$$ A ( s) = \int\limits _ { M } L ( x , u ( x) , u _ {, \alpha } ( x) ) \omega . $$

An extremal $ u ( x) $ for this functional (in a problem with fixed ends) satisfies the Euler–Lagrange equations

$$ \frac{\delta L }{\delta u ^ {a} } \equiv \ \sum _ {\begin{array}{c} \alpha = ( \alpha _ {1} \dots \alpha _ {n} ) \\ | \alpha | \leq k \end{array} } ( - 1 ) ^ {| \alpha | } \frac{d ^ \alpha }{d x ^ \alpha } \frac{\partial L }{\partial u _ {, \alpha } ^ {a} } = 0 , $$

where

$$ \frac{d ^ \alpha }{d x ^ \alpha } = \ \left ( \frac{d}{d x ^ {1} } \right ) ^ {\alpha _ {1} } \dots \left ( \frac{d}{d x ^ {n} } \right ) ^ {\alpha _ {n} } $$

are the total derivatives. An infinitesimal automorphism of $ \pi $, that is, a vector field $ Z $ on $ E $ of the form

$$ Z = X ^ {i} ( x) \frac \partial {\partial x ^ {i} } + U ^ {a} ( x , u ) \frac \partial {\partial u ^ {a} } , $$

is called an infinitesimal symmetry of $ A $ if the Lie derivative of the Lagrange $ n $- form $ L \omega \in \Lambda ^ {n} ( J ^ {k} E ) $ in the direction of the vector field $ Z ^ {(} k) $, which is generated by $ Z $ on $ J ^ {k} E $, vanishes:

$$ Z ^ {(} k) ( L \omega ) = 0 . $$

For the Lie derivative the following fundamental Noether formula holds:

$$ Z ^ {(} k) ( L \omega ) = \ \left [ {\overline{U}\; } {} ^ {a} \frac{\delta L }{\delta u ^ {a} } + \frac{d}{d x ^ {i} } J ^ {i} \right ] \omega , $$

where

$$ {\overline{U}\; } {} ^ {a} = \ U ^ {a} - u _ {,i} ^ {a} X ^ {i} ,\ J ^ {i} = \ L X ^ {i} + F ^ { i } , $$

and the $ F ^ { i } $ are the components of a certain vector field depending on $ {\overline{U}\; } {} ^ {a} $, $ L $ and their derivatives. In particular, $ F ^ { i } = {\overline{U}\; } {} ^ {a} ( \partial L / \partial u _ {,i} ^ {a} ) $ for $ k = 1 $. If $ Z $ is an infinitesimal symmetry, then

$$ - {\overline{U}\; } {} ^ {a} \frac{\delta L }{\delta u ^ {a} } = \ \frac{d}{d x ^ {i} } J ^ {i} , $$

that is, a certain linear combination of the variational derivatives $ \delta L / \delta u ^ {a} $ of the Lagrange function $ L $ is the divergence of the vector field $ J = J ^ {i} \partial / \partial x ^ {i} $. It is in this form that E. Noether stated her first theorem. The divergence of $ J $( a so-called Noether current) vanishes on extremals of the action functional, and the $ ( n - 1 ) $- form $ v _ {z} = J \llcorner \omega $ dual to it, which is obtained from $ \omega $ by inner multiplication by $ J $, is closed, that is, it is a conservation law.

There are important generalizations of Noether's theorem (see, for example, –). They are based on an extension of the concept of an infinitesimal symmetry. Instead of vector fields on $ E $ to which correspond one-parameter groups of transformations one considers vector fields on $ E $ with coefficients depending on the sections $ U ( x) $ and their derivatives of arbitrary order. Such fields $ Y $ no longer determine one-parameter transformation groups; however, one can define for them by purely algebraic means the concept of a Lie derivative. A field $ Y $ is called an algebraic infinitesimal symmetry if the Lie derivative of the Lagrange form vanishes in the direction of this field (maybe after restricting to extremals of the action functional). The generalized Noether theorem associates a conservation law with every algebraic symmetry. When applied to various equations of mathematical physics one obtains a large number of new important conservation laws.

Noether's second theorem asserts that if the action functional admits an infinite-dimensional Lie algebra of infinitesimal symmetries whose coefficients depend linearly on $ p $ arbitrary functions $ \phi ^ {1} ( x) \dots \phi ^ {p} ( x) $ and their derivatives up to order $ n $, then the variational derivatives $ \delta L / \delta u ^ {a} $ of the Lagrange function $ L $ satisfy a system of $ p $ differential equations of order $ m $. Namely, if

$$ Z = \phi ^ {s} U _ {s} ^ {a} \frac \partial {\partial u ^ {a} } + \phi _ {, \sigma } ^ {s} ( x) U _ {s} ^ {\sigma a } \frac \partial {\partial u ^ {a} } , $$

where

$$ \sigma = ( \sigma _ {1} \dots \sigma _ {n} ) ,\ \ | \sigma | = \sigma _ {1} + \dots + \sigma _ {n} \leq k , $$

is an infinitesimal symmetry for any smooth functions $ \phi ^ {s} ( x) $, $ s = 1 \dots p $, then identically

$$ U _ {s} ^ {a} \frac{\delta L }{\delta u ^ {a} } + ( - 1 ) ^ {| \sigma | } \frac{d ^ \sigma }{d x ^ \sigma } \left ( U _ {s} ^ {\sigma a } \frac{\delta L }{\delta u ^ {a} } \right ) = 0 ,\ \ s = 1 \dots p . $$

This theorem has applications, for example, in the theory of gauge fields.

Noether proved her first and second theorem in 1918 (see ).

References

[1a]	E. Noether, "Invarianten beliebiger Differentialausdrücke" Nachr. Gesellschaft. Wiss. Göttingen (1918) pp. 37–44; 240 (Also: Gesammelte Abh., Springer, 1983, pp. 240–247)
[1b]	E. Noether, "Invariante Variationsproblem" Nachr. Gesellschaft. Wiss. Göttingen (1918) pp. 237–257 (Also: Gesammelte Abh., Springer, 1983, pp. 248–270)
[2]	N.N. Bogolyubov, D.V. Shirkov, "Introduction to the theory of quantized fields" , Wiley (1980) (Translated from Russian)
[3]	I.M. Gel'fand, S.V. Fomin, "Calculus of variations" , Prentice-Hall (1963) (Translated from Russian)
[4]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[5]	L.V. [L.V. Ovsyannikov] Ovsiannikov, "Group analysis of differential equations" , Acad. Press (1982) (Translated from Russian)
[6]	Yu.I. Manin, "Algebraic aspects of nonlinear differential equations" J. Soviet Math. , 11 : 1 (1979) pp. 1–22 Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 11 (1978) pp. 5–152
[7]	A.M. Vinogradov, "On the algebro-geometric foundations of Lagrangian field theory" Soviet Math. Dokl. , 18 : 5 (1977) pp. 1200–1204 Dokl. Akad. Nauk SSSR , 236 : 2 (1977) pp. 284–287
[8]	V.V. Lychagin, "Contact geometry and non-linear second-order differential equations" Russian Math. Surveys , 34 : 1 (1979) pp. 149–180 Uspekhi Mat. Nauk , 34 : 1 (1979) pp. 137–165

References

[a1]	P.J. Olver, "Applications of Lie groups to differential equations" , Springer (1986)
[a2]	P. Funk, "Variationsrechnung und ihre Anwendung in Physik und Technik" , Springer (1962)
[a3]	W. Ludwig, C. Falter, "Symmetries in physics" , Springer (1988)
[a4]	T.-P. Cheng, L.-F. Li, "Gauge theory of elementary particle physics" , Oxford (1984)
[a5]	K. Uhlenbeck, "Conservation laws and their application in global differential geometry" B. Srinivasan (ed.) J. Sally (ed.) , Emmy Noether in Bryn Mawr , Springer (1983) pp. 103–117

Noether's normalization theorem: In any finitely-generated commutative integral $ k $- algebra $ A $ of transcendence degree $ d $ over a field $ k $ there are $ d $ elements $ x _ {1} \dots x _ {d} $ such that $ A $ is integral over the subalgebra $ B $ generated by them (cf. Integral ring; Integral extension of a ring). If $ A $ has a grading of the form $ A = \oplus _ {i \geq 0 } A _ {i} $, $ A _ {0} = k $, then $ x _ {1} \dots x _ {d} $ can be chosen to be homogeneous.

This theorem (sometimes also called Noether's normalization lemma) was proved by E. Noether [1]; in the graded case it was already stated by D. Hilbert [2].

The elements $ x _ {1} \dots x _ {d} $ are algebraically independent over $ k $, so that $ B $ is a polynomial algebra in these variables with coefficients in $ k $. If $ k $ is infinite, then $ x _ {1} \dots x _ {d} $ can be chosen from linear combinations of generators of $ A $ over $ k $. If $ k $ is algebraically closed, then the normalization theorem can be stated geometrically: Every irreducible affine $ d $- dimensional algebraic variety $ X $ is a finitely-sheeted (ramified) covering of an affine $ d $- dimensional space $ A ^ {d} $; more accurately, it has a finite morphism onto $ A ^ {d} $. Furthermore, if $ X $ is a closed subset of $ k ^ {n} $, then this morphism can be realized as the restriction to $ X $ of a certain linear mapping of $ k ^ {n} $ onto a $ d $- dimensional linear subspace.

The algebra $ A $ is finitely generated as a $ B $- module. The subalgebra $ B $ is not unique; however, a number of properties of $ A $ as a $ B $- module do not depend on the choice of $ B $. For example, if $ A $ is graded, as above under the hypotheses of the theorem, and if $ x _ {1} \dots x _ {d} $ are homogeneous (so that $ B $ is also graded), then the property of $ A $ of being a free $ B $- module does not depend on the choice of $ B $.

References

[1]	E. Noether, "Abstrakter Aufbau der Idealtheorie in algebraischen Zahl und Funktionenkörpern" Math. Ann. , 96 (1927) pp. 26–61
[2]	D. Hilbert, "Ueber die vollen Invariantensysteme" Math. Ann. , 42 (1893) pp. 313–373
[3]	M. Atiyah, I.G. Macdonald, "Introduction to commutative algebra" , Addison-Wesley (1969)
[4]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
[5]	O. Zariski, P. Samuel, "Commutative algebra" , 1–2 , v. Nostrand (1958–1960) ((reprinted: Springer, 1975))
[6]	S. Lang, "Algebra" , Addison-Wesley (1974)

V.L. Popov