Noether's theorem, proven by Emmy Noether in 1915, and published in 1918,^[1] states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves The action of a physical system is an integral of a so-called Lagrangian function, from which the system's behavior can be determined by the principle of least action. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system This article discusses the history of the principle of least action

Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world Calculus of variations is a field of Mathematics that deals with functionals, as opposed to ordinary Calculus which deals with functions. Noether's theorem allows a far-reaching generalization of earlier work on constants of motion in Lagrangian and Hamiltonian mechanics. In Mechanics, a constant of motion is a quantity that is conserved throughout the motion imposing in effect a constraint on the motion Lagrangian mechanics is a re-formulation of Classical mechanics that combines Conservation of momentum with Conservation of energy. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. However, Noether's theorem does not apply to systems that cannot be modeled with a Lagrangian; for example, dissipative systems with continuous symmetries need not have a corresponding conservation law. In Physics, dissipation embodies the concept of a Dynamical system where important mechanical modes such as Waves or Oscillations lose Energy

For illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry - it is the laws of motion which are symmetric. As another example, if a physical experiment has the same outcome regardless of place or time (having the same outcome, say, in Cleveland on Tuesday and Samaria on Wednesday), then its Lagrangian is symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively. In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός

Noether's theorem is profoundly important, both because of the insight it gives into conservation laws, but also as a practical calculational tool. It allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians to describe a physical system. For illustration, suppose that a new field is discovered that conserves a quantity X. Using Noether's theorem, the types of Lagrangians that conserve X because of a continuous symmetry can be determined, and then their fitness judged by other criteria.

There are numerous different versions of Noether's theorem, with varying degrees of generality. The original version only applied to ordinary differential equations (particles) and not partial differential equations (fields). In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i The original versions also assume that the Lagrangian only depends upon the first derivative, while later versions generalize the theorem to Lagrangians depending on the n^th derivative. There is also a quantum version of this theorem, known as the Ward-Takahashi identity. In Quantum field theory, a Ward-Takahashi identity is an identity between Correlation functions that follows from the global or gauged symmetries of the Generalizations of Noether's theorem to superspaces also exist. " Superspace " has had two meanings in physics The word was first used by John Wheeler to describe the Configuration space of General relativity; for example

Contents 1 Historical context 2 Mathematical expression 2.1 Field-theory version 3 Mathematical statement of the theorem 3.1 Explanation 4 Applications 5 Proof 5.1 Comments 5.2 Generalization to Lie algebras 5.3 Generalization of the proof 6 Examples 6.1 Example 1: Conservation of energy 6.2 Example 2: Conservation of center of momentum 6.3 Example 3: Conformal transformation 7 See also 8 References 9 Bibliography 10 External links

Historical context

Main articles: Constant of motion, conservation law, and conserved current

A conservation law states that some quantity X describing a system remains constant throughout its motion; expressed mathematically, the rate of change of X (its derivative with respect to time) is zero

Such quantities—often called constants of motion, although motion per se need not be involved, just evolution in time—are said to be conserved. In Mechanics, a constant of motion is a quantity that is conserved throughout the motion imposing in effect a constraint on the motion In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves In Physics, a conserved current J is the flow of the Canonical conjugate of a quantity possessing a Continuous Translational In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of In Mechanics, a constant of motion is a quantity that is conserved throughout the motion imposing in effect a constraint on the motion For example, if the energy of a system is conserved, its energy is constant at all times, which imposes a constraint on the system's motion and may help to solve for it. Aside from the insight that such constants of motion throw on the nature of system, they are a useful calculational tool; for example, an approximate solution can be corrected by finding the nearest state that satisfies the necessary conservation laws.

The earliest constants of motion discovered were momentum and energy, which were proposed in the 17th century by René Descartes and Gottfried Leibniz on the basis of collision experiments, and refined by subsequent researchers. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός A collision is an isolated event in which two or more bodies (colliding bodies exert relatively strong forces on each other for a relatively short time Isaac Newton was the first to enunciate the conservation of momentum in its modern form, and showed that it was a consequence of Newton's third law; interestingly, conservation of momentum still holds even in situations when Newton's third law is incorrect. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Modern physics has revealed that the conservation laws of momentum and energy are only approximately true, but their modern refinements—the conservation of four-momentum in special relativity and the zero divergence of the stress-energy tensor in general relativity—are rigorously true within the limits of those theories. In Special relativity, four-momentum is the generalization of the classical three-dimensional Momentum to four-dimensional Spacetime. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the The stress-energy tensor (sometimes stress-energy-momentum tensor is a Tensor quantity in Physics that describes the Density and Flux General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 The conservation of angular momentum, a generalization to rotating rigid bodies, likewise holds in modern physics. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position Another important conserved quantity, discovered in studies of the celestial mechanics of astronomical bodies, was the Laplace-Runge-Lenz vector. Celestial mechanics is the branch of Astrophysics that deals with the motions of Celestial objects The field applies principles of Physics, historically

In the late 18th and early 19th centuries, physicists developed more systematic methods for discovering conserved quantities. A major advance came in 1788 with the development of Lagrangian mechanics, which is related to the principle of least action. Lagrangian mechanics is a re-formulation of Classical mechanics that combines Conservation of momentum with Conservation of energy. This article discusses the history of the principle of least action In this approach, the state of the system can be described by any type of generalized coordinates q; the laws of motion need not be expressed in a Cartesian coordinate system, as was customary in Newtonian mechanics. By deriving Equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any Coordinate system that is ultimately In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane The action is defined as the time integral I of a function known as the Lagrangian L

where the dot over q signifies the rate of change of the coordinates q

Hamilton's principle states that the physical path q(t)—the one truly taken by the system—is a path for which infinitesimal variations in that path cause no change in I, at least up to first order. In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system In Physics, Hamilton's principle is William Rowan Hamilton 's formulation of the Principle of stationary action (see that article for historical formulations This principle results in the Euler–Lagrange equations

Thus, if one of the coordinates, say q_k, does not appear in the Lagrangian, the right-hand side of the equation is zero, and the left-hand side shows that

where the conserved momentum p_k is defined as the left-hand quantity in parentheses. In Calculus of variations, the Euler–Lagrange equation, or Lagrange's equation is a Differential equation whose solutions are the functions The absence of the coordinate q_k from the Lagrangian implies that the Langrangian is unaffected by changes or transformations of q_k; the Lagrangian is invariant, and is said to exhibit a kind of symmetry. Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these This is the seed idea from which Noether's theorem was born.

Several alternative methods for finding conserved quantities were developed in the 19th century, especially by William Rowan Hamilton. Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who For example, he developed a theory of canonical transformations that allowed researchers to change coordinates so that coordinates disappeared from the Lagrangian, resulting in a conserved quantities. In Hamiltonian mechanics, a canonical transformation is a change of Canonical coordinates (\mathbf{q} \mathbf{p} t \rightarrow (\mathbf{Q} \mathbf{P} t Another approach and perhaps the most efficient for finding conserved quantities is the Hamilton-Jacobi equation. In Physics, the Hamilton–Jacobi equation (HJE is a reformulation of Classical mechanics and thus equivalent to other formulations such as Newton's laws of

Mathematical expression

The essence of Noether's theorem is the following. Imagine that the action I defined above is invariant under small perturbations (warpings) of the time variable t and the generalized coordinates q

where the perturbations δt and δq are both small but variable. By deriving Equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any Coordinate system that is ultimately For generality, assume that there might be several such symmetry transformations of the action, say, N; we may use an index r=1, 2, 3,. Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these . . ,N to keep track of them. Then a generic perturbation can be written as a linear sum of the individual types of perturbations

Using these definitions, Emmy Noether showed that the N quantities

are conserved, i. Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and e. , are constants of motion; this is a simple version of Noether's theorem. In Mechanics, a constant of motion is a quantity that is conserved throughout the motion imposing in effect a constraint on the motion

For illustration, consider a Lagrangian that does not depend on time, i. e. , that is invariant (symmetric) under changes t → t + δt, without any change in the coordinates q. In this case, N=1, T=1 and Q = 0; the corresponding conserved quantity is the total energy H^[2]

Similarly, consider a Lagrangian that does not depend on a coordinate q_k, i. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός e. , that is invariant (symmetric) under changes q_k → q_k + δq_k. In that case, N=1, T = 0, and Q_k=1; the conserved quantity is the corresponding momentum p_k^[3]

In special and general relativity, these apparently separate conservation laws are aspects of a single conservation law, that of the stress-energy tensor,^[4] which is derived in the next section. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 The stress-energy tensor (sometimes stress-energy-momentum tensor is a Tensor quantity in Physics that describes the Density and Flux

The conservation of the angular momentum L = r × p is slightly more complicated to derive, but analgous to its linear momentum counterpart. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position ^[5] It is assumed that the symmetry of the Lagrangian is rotational, i. e. , that the Lagrangian does not depend on the absolute orientation of the physical system in space. For concreteness, assume that the Lagrangian does not change under small rotations of an angle δθ about an axis n; such a rotation transforms the Cartesian coordinates by the equation

Since time is not being transformed, T equals zero. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane Taking δθ as the ε parameter and the Cartesian coordinates r as the generalized coordinates q, the corresponding Q variables are given by

Then Noether's theorem states that the following quantity is conserved

In other words, the component of the angular momentum L along the n axis is conserved. If n is arbitrary, i. e. , if the system is insensitive to any rotation, then every component of L is conserved; in short, angular momentum is conserved. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position

Field-theory version

Mathematicians strive for the greatest generality and Emmy Noether was no exception. Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and Although useful in its own right, the version of her theorem just given was a special case the general version she derived in 1915. To give the flavor of the general theorem, a version of the Noether theorem for continuous fields in four-dimensional space-time is now given. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Since field theory problems are more common in modern physics than mechanics problems, this field-theory version is the most commonly used version of Noether's theorem. Mechanics ( Greek) is the branch of Physics concerned with the behaviour of physical bodies when subjected to Forces or displacements

Let there be a set of differentiable fields φ_k defined over all space and time; for example, the temperature T(x, t) would be representative of such a field, being a number defined at every place and time. The principle of least action can be applied to such fields, but the action is now an integral over space and time

(the theorem can actually be further generalized to the case where the Lagrangian depends on up to the n^th derivative using jet bundles)

Let the action be invariant under certain transformations of the space-time coordinates x^μ and the fields φ_k

where the transformations can be indexed by r=1, 2, 3,. This article discusses the history of the principle of least action In Differential geometry, the jet bundle is a certain construction which makes a new smooth Fiber bundle out of a given smooth fiber bundle . . ,N

For such systems, Noether's theorem states that there are N conserved current densities

In such cases, the conservation law is expressed in a four-dimensional way

which expresses the idea that the amount of a conserved quantity within a sphere cannot change unless some of it flows out of the sphere. In Physics, a conserved current J is the flow of the Canonical conjugate of a quantity possessing a Continuous Translational In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves For example, electric charge is conserved; the amount of charge within a sphere cannot change unless some of the charge leaves the sphere. Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction.

For illustration, consider a physical system of fields that behaves the same under translations in time and space, as considered above; in other words, the fields do not depend on the absolute position in space and time. In that case, N=4, one for each dimension of space and time. Since only the positions in space-time are being warped, not the fields, the Ψ are all zero and the X_μ^ν equal the Kronecker delta δ_μ^ν, where we have used μ instead of r for the index. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two In that case, Noether's theorem corresponds to the conservation law for the stress-energy tensor T_μ^ν[4]

The conservation of electric charge can be derived by considering transformations of the fields themselves. The stress-energy tensor (sometimes stress-energy-momentum tensor is a Tensor quantity in Physics that describes the Density and Flux Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. ^[6] In quantum mechanics, the probability amplitude ψ(x) of finding a particle at a point x is a complex field, because it ascribes a complex number to every point in space and time. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Quantum mechanics, a probability amplitude is a complex -valued function that describes an uncertain or unknown quantity Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted The probability amplitude itself is physically unmeasurable; only the probability p = |ψ|² is directly measureable. Therefore, the system is invariant under transformations of the ψ field and its complex conjugate field ψ^* that leave |ψ|² unchanged, such as

In the limit when θ becomes infinitesimally small (δθ), it may be taken as the ε, and the Ψ are equal to iψ and -iψ^*, respectively. In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. A specific example is the Klein-Gordon equation, the relativistically correct version of the Schrödinger equation for spinless particles, which has the Lagrangian density

In this case, Noether's theorem states that the conserved current equals

which, when multiplied by the charge, equals the electric current density. The Klein–Gordon equation ( Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin This transformation was first noted by Hermann Weyl and is one of the fundamental gauge symmetries of modern physics. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations

Mathematical statement of the theorem

Informally, Noether's theorem can be stated as (technical fine points aside):

To every differentiable symmetry generated by local actions, there corresponds a conserved current. Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these In Physics, a conserved current J is the flow of the Canonical conjugate of a quantity possessing a Continuous Translational

Explanation

The word "symmetry" in the above statement refers more precisely to the covariance of the form that a physical law takes with respect to a one-dimensional Lie group of transformations satisfying certain technical criteria. In Physics, a covariant transformation is a rule (specified below that describes how certain physical entities change under a change of Coordinate system. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group The conservation law of a physical quantity is usually expressed as a continuity equation. In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves A physical Quantity is a physical property that can be quantified A continuity equation is a Differential equation that describes the conservative transport of some kind of quantity

The formal proof of the theorem uses only the condition of invariance to derive an expression for a current associated with a conserved physical quantity. The conserved quantity is called the Noether charge and the flow carrying that 'charge' is called the Noether current. The Noether current is defined up to a solenoidal vector field. In Vector calculus a solenoidal vector field (also known as an incompressible vector field) is a Vector field v with Divergence zero

Applications

Application of Noether's theorem allows physicists to gain powerful insights into any general theory in physics, by just analyzing the various transformations that would make the form of the laws involved invariant. For example:

• the invariance of physical systems with respect to spatial translation (in other words, that the laws of physics do not vary with locations in space) gives the law of conservation of linear momentum;
• invariance with respect to rotation gives the law of conservation of angular momentum;
• invariance with respect to time translation gives the well known law of conservation of energy

In quantum field theory, the analog to Noether's theorem, the Ward-Takahashi identities, yields further conservation laws, such as the conservation of electric charge from the invariance with respect to a change in the phase factor of the complex field of the charged particle and the associated gauge of the electric potential and vector potential. In Physics, translation is movement that changes the position of an object as opposed to Rotation. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of In Physics, the law of conservation of energy states that the total amount of Energy in an isolated system remains constant and cannot be created although it may In quantum field theory (QFT the forces between particles are mediated by other particles In Quantum field theory, a Ward-Takahashi identity is an identity between Correlation functions that follows from the global or gauged symmetries of the Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Quantum Mechanics, a phase factor is a complex scalar number of Absolute value 1 that multiplies a bra or ket. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field In Vector calculus, a vector potential is a Vector field whose curl is a given vector field

The Noether charge is also used in calculating the entropy of stationary black holes^[7]. In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside

Proof

Suppose we have an n-dimensional manifold, M and a target manifold T. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Let be the configuration space of smooth functions from M to T. "Configuration space" may also refer to PCI Configuration Space. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability (More generally, we can have smooth sections of a fiber bundle over M)

Examples of this "M" in physics include:

• In classical mechanics, in the Hamiltonian formulation, M is the one-dimensional manifold R, representing time and the target space is the cotangent bundle of space of generalized positions. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. In Mathematics, especially Differential geometry, the cotangent bundle of a Smooth manifold is the Vector bundle of all the Cotangent spaces Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another
• In field theory, M is the spacetime manifold and the target space is the set of values the fields can take at any given point. In Physics, a field is a Physical quantity associated to each point of Spacetime. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS For example, if there are m real-valued scalar fields, φ₁,. In Mathematics, the real numbers may be described informally in several different ways In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point . . ,φ_m, then the target manifold is R^m. If the field is a real vector field, then the target manifold is isomorphic to R³. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

Now suppose there is a functional

called the action. In Mathematics, a functional is traditionally a map from a Vector space to the field underlying the vector space which is usually the Real In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation (Note that it takes values into , rather than ; this is for physical reasons, and doesn't really matter for this proof. )

To get to the usual version of Noether's theorem, we need additional restrictions on the action. In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation We assume is the integral over M of a function

called the Lagrangian, depending on φ, its derivative and the position. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In other words, for φ in

Suppose we are given boundary conditions, ie. In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints , a specification of the value of at the boundary if M is compact, or some limit on as x approaches ∞. For a different notion of boundary related to Manifolds see that article Then the subspace of consisting of functions such that all functional derivatives of at φ are zero, that is:

and that satisfies the given boundary conditions, is the subspace of on shell solutions. Subspace may refer to;Mathematics Euclidean subspace, in linear algebra a set of vectors in n -dimensional Euclidean space that is closed under addition In Mathematics and theoretical Physics, the functional derivative is a generalization of the Directional derivative. In Physics, particularly in Quantum field theory, configurations of a physical system that satisfy classical Equations of motion are called on shell (See principle of stationary action)

Now, suppose we have an infinitesimal transformation on , generated by a functional derivation, Q such that

for all compact submanifolds N or in other words,

for all x, where we set . This article discusses the history of the principle of least action In Mathematics, an infinitesimal transformation is a limiting form of small transformation. In Mathematics, a functional is traditionally a map from a Vector space to the field underlying the vector space which is usually the Real

If this holds on shell and off shell, we say Q generates an off-shell symmetry. In Physics, particularly in Quantum field theory, configurations of a physical system that satisfy classical Equations of motion are called on shell In Physics, particularly in Quantum field theory, configurations of a physical system that satisfy classical Equations of motion are called on shell If this only holds on shell, we say Q generates an on-shell symmetry. In Physics, particularly in Quantum field theory, configurations of a physical system that satisfy classical Equations of motion are called on shell Then, we say Q is a generator of a one parameter symmetry Lie group. In Mathematics, a one-parameter group or one-parameter subgroup usually means a continuous Group homomorphism φ: R Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

Now, for any N, because of the Euler–Lagrange theorem, on shell (and only on-shell), we have

Since this is true for any N, we have

But this is the continuity equation for the current defined by

which is called the Noether current associated with the symmetry. In Calculus of variations, the Euler–Lagrange equation, or Lagrange's equation is a Differential equation whose solutions are the functions In Physics, particularly in Quantum field theory, configurations of a physical system that satisfy classical Equations of motion are called on shell A continuity equation is a Differential equation that describes the conservative transport of some kind of quantity Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or The continuity equation tells us that if we integrate this current over a space-like slice, we get a conserved quantity called the Noether charge (provided, of course, if M is noncompact, the currents fall off sufficiently fast at infinity). The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves

Comments

Noether's theorem is really a reflection of the relation between the boundary conditions and the variational principle. Assuming no boundary terms in the action, Noether's theorem implies that

Noether's theorem is an on shell theorem. In Physics, particularly in Quantum field theory, configurations of a physical system that satisfy classical Equations of motion are called on shell The quantum analog of Noether's theorem are the Ward-Takahashi identities. In Quantum field theory, a Ward-Takahashi identity is an identity between Correlation functions that follows from the global or gauged symmetries of the

Generalization to Lie algebras

Suppose say we have two symmetry derivations Q₁ and Q₂. Then, [Q₁,Q₂] is also a symmetry derivation. Let's see this explicitly. Let's say

and

Then,

where f₁₂=Q₁[f₂^μ]-Q₂[f₁^μ]. So,

This shows we can (trivially) extend Noether's theorem to larger Lie algebras.

Generalization of the proof

This applies to any local symmetry derivation Q satisfying , and also to more general local functional differentiable actions, including ones where the Lagrangian depends on higher derivatives of the fields. Let ε be any arbitrary smooth function of the spacetime (or time) manifold such that the closure of its support is disjoint from the boundary. ε is a test function. In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions Then, because of the variational principle (which does not apply to the boundary, by the way), the derivation distribution q generated by q[ε][Φ(x)]=ε(x)Q[Φ(x)] satisfies for any ε, or more compactly, for all x not on the boundary (but remember that q(x) is a shorthand for a derivation distribution, not a derivation parametrized by x in general). This is the generalization of Noether's theorem.

To see how the generalization related to the version given above, assume that the action is the spacetime integral of a Lagrangian which only depends on and its first derivatives. Also, assume

Then,

for all ε.

More generally, if the Lagrangian depends on higher derivatives, then

Examples

Example 1: Conservation of energy

Looking at the specific case of a Newtonian particle of mass m, coordinate x, moving under the influence of a potential V, coordinatized by time t. The action, S, is:

Consider the generator of time translations . In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation In other words, . (Quantum field theoreticians would often put a factor of i on the right hand side. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation ) Note that x has an explicit dependence on time, whilst V does not; consequently:

so we can set

Then,

The right hand side is the energy and Noether's theorem states that (i. e. the principle of conservation of energy is a consequence of invariance under time translations).

More generally, if the Lagrangian does not depend explicitly on time, the quantity

(called the Hamiltonian) is conserved. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.

Example 2: Conservation of center of momentum

Still considering 1-dimensional time, let

i. e. N Newtonian particles where the potential only depends pairwise upon the relative displacement.

For , let's consider the generator of Galilean transformations (i. e. a change in the frame of reference). In other words,

Note that

This has the form of so we can set

Then,

where is the total momentum, M is the total mass and is the center of mass. Noether's theorem states:

.

Example 3: Conformal transformation

Both examples 1 and 2 are over a 1-dimensional manifold (time). For an example involving spacetime, let's work out the case of a conformal transformation of a massless real scalar field with a quartic potential in (3 + 1)-Minkowski spacetime. In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity

For Q, let's consider the generator of a spacetime rescaling. In other words,

The second term on the right hand side is due to the "conformal weight" of φ. Note that

This has the form of

(where we have performed a change of dummy indices) so we can set

Then,

Noether's theorem states that (as one may explicitly check by substituting the Euler-Lagrange equations into the left hand side).

(Aside: If you try to find the Ward-Takahashi analog of this equation, you'd run into a problem because of anomalies. In Quantum field theory, a Ward-Takahashi identity is an identity between Correlation functions that follows from the global or gauged symmetries of the In Quantum physics an anomaly or quantum anomaly is the failure of a Symmetry of a theory's classical action to be a symmetry of any regularization )

See also

• Charge (physics)

• Invariant (physics)

References

Bibliography

• Lanczos C (1970). Cornelius Lanczos ( Lánczos Kornél) (approximate pronunciation 'LAHNT-sawsh') born Löwy Kornél ( February 2, 1893 The variational Principles of Mechanics, 4th edition, New York: Dover Publications, pp. 401–405. ISBN 0-486-65067-7.  

• Goldstein H (1980). Herbert Goldstein ( June 26, 1922 &ndash January 12, 2005) was an American Physicist and the author of the standard graduate Classical Mechanics, 2nd edition, Reading, MA: Addison-Wesley Publishing, pp. Classical Mechanics is a Textbook about Classical Mechanics written by Herbert Goldstein. 588–596. ISBN 0-201-02918-9.  

External links

• English translation of Noether's paper
• Article on Noether's theorem by John Baez
• E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws by Nina Byers
• Noether's Theorem at MathPages
• Symmetric energy-momentum tensor in Maxwell, Yang-Mills, and Proca theories obtained using only Noether's theorem

John Carlos Baez (born 1961 is an American mathematical physicist at the University of California Riverside. Nina Byers is Research Professor and Professor of Physics Emeritus at UCLA.

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