Noether's second theorem
Encyclopedia
In mathematics
, Noether's second theorem relates symmetries of an action
functional
with a system of differential equation
s. 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 derivative
s 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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Noether's second theorem relates symmetries of an action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
functional
Functional
Generally, functional refers to something able to fulfill its purpose or function.*Functionalism and Functional form, movements in architectural design*Functional group, certain atomic combinations that occur in various molecules, e.g...
with a system of differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s. The action S of a physical system is an integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of a so-called Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
function L, from which the system's behavior can be determined by the principle of least action
Principle of least action
In physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system...
.
Specifically, the theorem says that if the action has an infinite-dimensional Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
of infinitesimal symmetries parameterized linearly by k arbitrary functions and their derivatives up to order m, then the functional derivative
Functional derivative
In mathematics and theoretical physics, the functional derivative is a generalization of the gradient. While the latter differentiates with respect to a vector with discrete components, the former differentiates with respect to a continuous function. Both of these can be viewed as extensions of...
s of L satisfy a system of k differential equations.
Noether's second theorem is sometimes used in gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
. Gauge theories are the basic elements of all modern field theories of physics, such as the prevailing Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
.
See also
- Noether's first theorem
- Noether identitiesNoether identitiesIn mathematics, Noether identities characterize the degeneracy of a Lagrangian system. Given a Lagrangian system and its Lagrangian L, Noether identities can be defined as a differential operator whose kernel contains a range of the Euler–Lagrange operator of L...
- Gauge symmetry (mathematics)Gauge symmetry (mathematics)In mathematics, any Lagrangian system generally admits gaugesymmetries, though it may happen that they are trivial. Intheoretical physics, the notion of gaugesymmetries depending on parameter functions is a cornerstone ofcontemporary field theory....
- Emmy NoetherEmmy NoetherAmalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
External links
- English translation of Noether's paper
- Fulp, R., Lada, T., Stasheff, J.. Noether variational theorem II and the BV formalism, arXiv: math/0204079
- Bashkirov, D., Giachetta, G., Mangiarotti, L., Sardanashvily, G.Gennadi SardanashvilyGennadi Sardanashvily is a theoretical physicist, a principal research scientist of Moscow State University.- Biography :...
, The KT-BRST complex of a degenerate Lagrangian system, Lett. Math. Phys. 83 (2008) 237; arXiv: math-ph/0702097.