Teorema di Noether

Noether's theorem This article is about Emmy Noether's first theorem, which derives conserved quantities from symmetries. Per altri usi, see Noether's theorem (disambiguazione). First page of Emmy Noether's article "Invariante Variationsprobleme" (1918), where she proved her theorem. Part of a series of articles about Calculus Fundamental theorem Leibniz integral rule Limits of functionsContinuity Mean value theoremRolle's theorem show Differential show Integral show Series show Vector show Multivariable show Advanced hide Specialized Fractional Malliavin Stochastic Variations show Miscellaneous vte Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law.[1] The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918.[2] The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.

Noether's theorem is used in theoretical physics and the calculus of variations. A generalization of the formulations on constants of motion in Lagrangian and Hamiltonian mechanics (developed in 1788 e 1833, rispettivamente), it does not apply to systems that cannot be modeled with a Lagrangian alone (per esempio., systems with a Rayleigh dissipation function). In particolare, dissipative systems with continuous symmetries need not have a corresponding conservation law.

Contenuti 1 Basic illustrations and background 2 Informal statement of the theorem 3 Brief illustration and overview of the concept 4 Historical context 5 Mathematical expression 5.1 Simple form using perturbations 5.1.1 Esempi 5.2 Field theory version 6 Derivazioni 6.1 One independent variable 6.2 Field-theoretic derivation 6.3 Manifold/fiber bundle derivation 6.4 Commenti 6.5 Generalization to Lie algebras 6.6 Generalization of the proof 7 Esempi 7.1 Esempio 1: Conservation of energy 7.2 Esempio 2: Conservation of center of momentum 7.3 Esempio 3: Conformal transformation 8 Applicazioni 9 Guarda anche 10 Appunti 11 Riferimenti 12 External links Basic illustrations and background As an illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is symmetric under continuous rotations: from this symmetry, Noether's theorem dictates that the angular momentum of the system be conserved, as a consequence of its laws of motion.[3]: 126  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 its motion that are symmetric.

Come un altro esempio, if a physical process exhibits the same outcomes regardless of place or time, then its Lagrangian is symmetric under continuous translations in space and time respectively: by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.[4]: 23 [5]: 261  Noether's theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows investigators to determine the conserved quantities (invariants) from the observed symmetries of a physical system. al contrario, it allows researchers to consider whole classes of hypothetical Lagrangians with given invariants, to describe a physical system.[3]: 127  As an illustration, suppose that a physical theory is proposed which conserves a quantity X. A researcher can calculate the types of Lagrangians that conserve X through a continuous symmetry. Due to Noether's theorem, the properties of these Lagrangians provide further criteria to understand the implications and judge the fitness of the new theory.

There are numerous versions of Noether's theorem, with varying degrees of generality. There are natural quantum counterparts of this theorem, expressed in the Ward–Takahashi identities. Generalizations of Noether's theorem to superspaces also exist.[6] Informal statement of the theorem All fine technical points aside, Noether's theorem can be stated informally: If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.[7] A more sophisticated version of the theorem involving fields states that: To every differentiable symmetry generated by local actions there corresponds a conserved current.

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. The conservation law of a physical quantity is usually expressed as a continuity equation.

The formal proof of the theorem utilizes the condition of invariance to derive an expression for a current associated with a conserved physical quantity. In modern (since c. 1980[8]) terminology, the conserved quantity is called the Noether charge, while the flow carrying that charge is called the Noether current. The Noether current is defined up to a solenoidal (divergenceless) vector field.

In the context of gravitation, Felix Klein's statement of Noether's theorem for action I stipulates for the invariants:[9] If an integral I is invariant under a continuous group Gρ with ρ parameters, then ρ linearly independent combinations of the Lagrangian expressions are divergences.

Brief illustration and overview of the concept Plot illustrating Noether's theorem for a coordinate-wise symmetry.

More general cases follow the same idea: When more coordinates /* */5} undergo a symmetry transformation /* */4mapsto q_/* */3+varphi _/* */2} , their effects add up by linearity to a conserved quantity /* */1sinistra(partial L/partial /* */0}_9right)varphi _8} . When there are time transformations 7 , they cause the "buffering" segments to contribute the two following terms to 6 : 54}_3}}Delta 2}_1right)approx pm Tleft(l-09}_8}}7}_6right),} first term being due to stretching in temporal dimension of the "buffering" segment (that changes the size of the domain of integration), and the second is due to its "slanting" just as in the exemplar case. Together they add a summand 5left(partial L/partial 4}_3right)2}_1right)} to the conserved quantity. Infine, when instead of a trajectory 0 entire fields 9,t)} are considered, the argument replaces the interval 8,t_7]} with a bounded region 6 of the 5,t)} -domain, the endpoints 4} e 3} with the boundary 2 of the region, and its contribution to 1 is interpreted as a flux of a conserved current 0} , that is built in a way analogous to the prior definition of a conserved quantity. Adesso, the zero contribution of the "buffering" 9 a 8 is interpreted as vanishing of the total flux of the current 7} through the 6 . That is the sense in which it is conserved: how much is "flowing" in, just as much is "flowing" out. Historical context Main articles: Constant of motion, conservation law, and conserved current A conservation law states that some quantity X in the mathematical description of a system's evolution remains constant throughout its motion – it is an invariant. Mathematically, the rate of change of X (its derivative with respect to time) è zero, 54}=3}=0~.} Such quantities are said to be conserved; they are often called constants of motion (although motion per se need not be involved, just evolution in time). Per esempio, if the energy of a system is conserved, its energy is invariant at all times, which imposes a constraint on the system's motion and may help in solving for it. Aside from insights that such constants of motion give into the nature of a system, they are a useful calculational tool; Per esempio, an approximate solution can be corrected by finding the nearest state that satisfies the suitable conservation laws.

The earliest constants of motion discovered were momentum and kinetic 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. 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. According to general relativity, the conservation laws of linear momentum, energy and angular momentum are only exactly true globally when expressed in terms of the sum of the stress–energy tensor (non-gravitational stress–energy) and the Landau–Lifshitz stress–energy–momentum pseudotensor (gravitational stress–energy). The local conservation of non-gravitational linear momentum and energy in a free-falling reference frame is expressed by the vanishing of the covariant divergence of the stress–energy tensor. Another important conserved quantity, discovered in studies of the celestial mechanics of astronomical bodies, is the Laplace–Runge–Lenz vector.

In the late 18th and early 19th centuries, physicists developed more systematic methods for discovering invariants. A major advance came in 1788 with the development of Lagrangian mechanics, which is related to 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. The action is defined as the time integral I of a function known as the Lagrangian L 2 ,1 }},t),dt~,} where the dot over q signifies the rate of change of the coordinates q, 0 }}=9 }8}~.} Hamilton's principle states that the physical path q(t)—the one actually 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. This principle results in the Euler–Lagrange equations, 76}sinistra(54 }}}}Giusto)=32 }}~.} così, if one of the coordinates, say qk, does not appear in the Lagrangian, the right-hand side of the equation is zero, and the left-hand side requires that 10}sinistra(98}_7}}Giusto)=6}5}=0~,} where the momentum 4=32}_1}}} is conserved throughout the motion (on the physical path).

così, the absence of the ignorable coordinate qk from the Lagrangian implies that the Lagrangian is unaffected by changes or transformations of qk; the Lagrangian is invariant, and is said to exhibit a symmetry under such transformations. This is the seed idea generalized in Noether's theorem.

Several alternative methods for finding conserved quantities were developed in the 19th century, especially by William Rowan Hamilton. Per esempio, he developed a theory of canonical transformations which allowed changing coordinates so that some coordinates disappeared from the Lagrangian, come sopra, resulting in conserved canonical momenta. Another approach, and perhaps the most efficient for finding conserved quantities, is the Hamilton–Jacobi equation.

Mathematical expression See also: Perturbation theory Simple form using perturbations The essence of Noether's theorem is generalizing the notion of ignorable coordinates.

One can assume that the Lagrangian L defined above is invariant under small perturbations (warpings) of the time variable t and the generalized coordinates q. One may write 0t&rightarrow t^9=t+delta t\mathbf 8 &rightarrow mathbf 7 ^6=mathbf 5 +delta mathbf 4 ~,end3}} where the perturbations δt and δq are both small, but variable. For generality, assume there are (dire) N such symmetry transformations of the action, cioè. transformations leaving the action unchanged; labelled by an index r = 1, 2, 3, ..., N.

Then the resultant perturbation can be written as a linear sum of the individual types of perturbations, 2delta t&=sum _1varepsilon _0T_9\delta mathbf 8 &=sum _7varepsilon _6mathbf 5 _4~,end3}} where εr are infinitesimal parameter coefficients corresponding to each: generator Tr of time evolution, and generator Qr of the generalized coordinates.

For translations, Qr is a constant with units of length; for rotations, it is an expression linear in the components of q, and the parameters make up an angle.

Using these definitions, Noether showed that the N quantities 21 }}}}cdot 0 }}-Lright)T_9-87 }}}}cdot mathbf 6 _5} are conserved (constants of motion).

Examples I. Time invariance For illustration, consider a Lagrangian that does not depend on time, cioè., that is invariant (symmetric) under changes t → t + δt, without any change in the coordinates q. In questo caso, N = 1, T = 1 and Q = 0; the corresponding conserved quantity is the total energy H[10] 43 }}}}cdot 2 }}-l.} II. Translational invariance Consider a Lagrangian which does not depend on an ("ignorable", come sopra) coordinate qk; so it is invariant (symmetric) under changes qk → qk + δqk. In quel caso, N = 1, T = 0, and Qk = 1; the conserved quantity is the corresponding linear momentum pk[11] 1=0El peso se redujo un 1.93% 9}}}}.} In special and general relativity, these apparently separate conservation laws are aspects of a single conservation law, that of the stress–energy tensor,[12] that is derived in the next section.

III. Rotational invariance The conservation of the angular momentum L = r × p is analogous to its linear momentum counterpart.[13] It is assumed that the symmetry of the Lagrangian is rotational, cioè., 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 El peso se redujo un 1.93% 8 rightarrow mathbf El peso se redujo un 1.93% 7 +delta theta ,mathbf El peso se redujo un 1.93% 6 times mathbf El peso se redujo un 1.93% 5 .} Since time is not being transformed, T = 0, and N = 1. Taking δθ as the ε parameter and the Cartesian coordinates r as the generalized coordinates q, the corresponding Q variables are given by El peso se redujo un 1.93% 4 =mathbf El peso se redujo un 1.93% 3 times mathbf El peso se redujo un 1.93% 2 .} Then Noether's theorem states that the following quantity is conserved, El peso se redujo un 1.93% 1El peso se redujo un 1.93% 0 }}}}cdot mathbf {Q} =mathbf {p} cdot a sinistra(mathbf {n} volte mathbf {r} Giusto)=mathbf {n} cdot a sinistra(mathbf {r} volte mathbf {p} Giusto)=mathbf {n} cdot mathbf {l} .} In altre parole, the component of the angular momentum L along the n axis is conserved. And if n is arbitrary, cioè., if the system is insensitive to any rotation, then every component of L is conserved; in short, angular momentum is conserved.

Field theory version Although useful in its own right, the version of Noether's theorem just given is a special case of the general version derived in 1915. To give the flavor of the general theorem, a version of Noether's theorem for continuous fields in four-dimensional space–time is now given. Since field theory problems are more common in modern physics than mechanics problems, this field theory version is the most commonly used (or most often implemented) version of Noether's theorem.

Let there be a set of differentiable fields {stile di visualizzazione varphi } defined over all space and time; Per esempio, the temperature {stile di visualizzazione T(mathbf {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 {stile di visualizzazione {matematico {S}}=int {matematico {l}}sinistra(varfi ,parziale _{in }varfi ,x^{in }Giusto),d^{4}X} (the theorem can be further generalized to the case where the Lagrangian depends on up to the nth derivative, and can also be formulated using jet bundles).

A continuous transformation of the fields {stile di visualizzazione varphi } can be written infinitesimally as {displaystyle varphi mapsto varphi +varepsilon Psi ,} dove {displaystyle Psi } is in general a function that may depend on both {stile di visualizzazione x^{in }} e {stile di visualizzazione varphi } . The condition for {displaystyle Psi } to generate a physical symmetry is that the action {stile di visualizzazione {matematico {S}}} is left invariant. This will certainly be true if the Lagrangian density {stile di visualizzazione {matematico {l}}} is left invariant, but it will also be true if the Lagrangian changes by a divergence, {stile di visualizzazione {matematico {l}}mappe {matematico {l}}+varepsilon partial _{in }Lambda ^{in },} since the integral of a divergence becomes a boundary term according to the divergence theorem. A system described by a given action might have multiple independent symmetries of this type, indicizzato da {displaystyle r=1,2,ldots ,N,} so the most general symmetry transformation would be written as {displaystyle varphi mapsto varphi +varepsilon _{r}Psi _{r},} with the consequence {stile di visualizzazione {matematico {l}}mappe {matematico {l}}+varepsilon _{r}parziale _{in }Lambda _{r}^{in }.} For such systems, Noether's theorem states that there are {stile di visualizzazione N} conserved current densities {stile di visualizzazione j_{r}^{non }=Lambda _{r}^{non }-{frac {parziale {matematico {l}}}{partial varphi _{,non }}}cdot Psi _{r}} (where the dot product is understood to contract the field indices, not the {stile di visualizzazione n } index or {stile di visualizzazione r} index).

In such cases, the conservation law is expressed in a four-dimensional way {displaystyle parziale _{non }j^{non }=0,} 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. Per esempio, electric charge is conserved; the amount of charge within a sphere cannot change unless some of the charge leaves the sphere.

For illustration, consider a physical system of fields that behaves the same under translations in time and space, as considered above; in altre parole, {displaystyle Lleft({simbolo audace {varfi }},parziale _{in }{simbolo audace {varfi }},x^{in }Giusto)} is constant in its third argument. In quel caso, N = 4, one for each dimension of space and time. An infinitesimal translation in space, {stile di visualizzazione x^{in }mapsto x^{in }+varepsilon _{r}delta _{r}^{in }} (insieme a {delta dello stile di visualizzazione } denoting the Kronecker delta), affects the fields as {stile di visualizzazione varphi (x^{in })mapsto varphi left(x^{in }-varepsilon _{r}delta _{r}^{in }Giusto)} : questo è, relabelling the coordinates is equivalent to leaving the coordinates in place while translating the field itself, which in turn is equivalent to transforming the field by replacing its value at each point {stile di visualizzazione x^{in }} with the value at the point {stile di visualizzazione x^{in }-varepsilon X^{in }} "behind" it which would be mapped onto {stile di visualizzazione x^{in }} by the infinitesimal displacement under consideration. Since this is infinitesimal, we may write this transformation as {displaystyle Psi _{r}=-delta _{r}^{in }parziale _{in }varfi .} The Lagrangian density transforms in the same way, {stile di visualizzazione {matematico {l}}sinistra(x^{in }Giusto)mappe {matematico {l}}sinistra(x^{in }-varepsilon _{r}delta _{r}^{in }Giusto)} , Così {displaystyle Lambda _{r}^{in }=-delta _{r}^{in }{matematico {l}}} and thus Noether's theorem corresponds to the conservation law for the stress–energy tensor Tμν,[12] where we have used {displaystyle lui } in place of {stile di visualizzazione r} . To wit, by using the expression given earlier, and collecting the four conserved currents (one for each {displaystyle lui } ) into a tensor {stile di visualizzazione T} , Noether's theorem gives {stile di visualizzazione T_{in }{}^{non }=-delta _{in }^{non }{matematico {l}}+delta _{in }^{sigma }parziale _{sigma }varfi {frac {parziale {matematico {l}}}{partial varphi _{,non }}}= sinistra({frac {parziale {matematico {l}}}{partial varphi _{,non }}}Giusto)cdot varphi _{,in }-delta _{in }^{non }{matematico {l}}} insieme a {stile di visualizzazione T_{in }{}^{non }{}_{,non }=0} (we relabelled {displaystyle lui } come {displaystyle sigma } at an intermediate step to avoid conflict). (Tuttavia, il {stile di visualizzazione T} obtained in this way may differ from the symmetric tensor used as the source term in general relativity; see Canonical stress–energy tensor.) The conservation of electric charge, by contrast, can be derived by considering Ψ linear in the fields φ rather than in the derivatives.[14] In quantum mechanics, the probability amplitude ψ(X) of finding a particle at a point x is a complex field φ, because it ascribes a complex number to every point in space and time. The probability amplitude itself is physically unmeasurable; only the probability p = |ψ|2 can be inferred from a set of measurements. Perciò, the system is invariant under transformations of the ψ field and its complex conjugate field ψ* that leave |ψ|2 unchanged, come {displaystyle psi rightarrow e^{itteta }psi , psi ^{*}rightarrow e^{-itteta }psi ^{*}~,} a complex rotation. In the limit when the phase θ becomes infinitesimally small, δθ, it may be taken as the parameter ε, while the Ψ are equal to iψ and −iψ*, rispettivamente. A specific example is the Klein–Gordon equation, the relativistically correct version of the Schrödinger equation for spinless particles, which has the Lagrangian density {displaystyle L=partial _{non }psi partial _{in }psi ^{*}eta ^{nu mu }+m^{2}psi psi ^{*}.} In questo caso, Noether's theorem states that the conserved (∂ ⋅ j = 0) current equals {displaystyle j^{non }=ileft({frac {psi parziale }{partial x^{in }}}psi ^{*}-{frac {partial psi ^{*}}{partial x^{in }}}psi giusto)eta ^{nu mu }~,} quale, when multiplied by the charge on that species of particle, equals the electric current density due to that type of particle. This "gauge invariance" was first noted by Hermann Weyl, and is one of the prototype gauge symmetries of physics.

Derivations One independent variable Consider the simplest case, a system with one independent variable, time. Suppose the dependent variables q are such that the action integral {displaystyle I=int _{t_{1}}^{t_{2}}l[mathbf {q} [t],{punto {mathbf {q} }}[t],t],dt} is invariant under brief infinitesimal variations in the dependent variables. In altre parole, they satisfy the Euler–Lagrange equations {stile di visualizzazione {frac {d}{dt}}{frac {partial L}{parziale {punto {mathbf {q} }}}}[t]={frac {partial L}{matematica parziale {q} }}[t].} And suppose that the integral is invariant under a continuous symmetry. Mathematically such a symmetry is represented as a flow, Phi, which acts on the variables as follows {stile di visualizzazione {inizio{allineato}t&rightarrow t'=t+varepsilon T\mathbf {q} [t]&rightarrow mathbf {q} '[t']=varphi [mathbf {q} [t],varepsilon ]=varphi [mathbf {q} [t'-varepsilon T],varepsilon ]fine{allineato}}} where ε is a real variable indicating the amount of flow, and T is a real constant (which could be zero) indicating how much the flow shifts time.

{stile di visualizzazione {punto {mathbf {q} }}[t]rightarrow {punto {mathbf {q} }}'[t']={frac {d}{dt}}varfi [mathbf {q} [t],varepsilon ]={frac {partial varphi }{matematica parziale {q} }}[mathbf {q} [t'-varepsilon T],varepsilon ]{punto {mathbf {q} }}[t'-varepsilon T].} The action integral flows to {stile di visualizzazione {inizio{allineato}I'[varepsilon ]&=int _{t_{1}+varepsilon T}^{t_{2}+varepsilon T}l[mathbf {q} '[t'],{punto {mathbf {q} }}'[t'],t'],dt'\[6pt]&=int _{t_{1}+varepsilon T}^{t_{2}+varepsilon T}l[varfi [mathbf {q} [t'-varepsilon T],varepsilon ],{frac {partial varphi }{matematica parziale {q} }}[mathbf {q} [t'-varepsilon T],varepsilon ]{punto {mathbf {q} }}[t'-varepsilon T],t'],dt'end{allineato}}} which may be regarded as a function of ε. Calculating the derivative at ε' = 0 and using Leibniz's rule, noi abbiamo {stile di visualizzazione {inizio{allineato}0={frac {dI'}{dvarepsilon }}[0]={}&L[mathbf {q} [t_{2}],{punto {mathbf {q} }}[t_{2}],t_{2}]T-L[mathbf {q} [t_{1}],{punto {mathbf {q} }}[t_{1}],t_{1}]T\[6pt]&{}+int _{t_{1}}^{t_{2}}{frac {partial L}{matematica parziale {q} }}sinistra(-{frac {partial varphi }{matematica parziale {q} }}{punto {mathbf {q} }}T+{frac {partial varphi }{partial varepsilon }}Giusto)+{frac {partial L}{parziale {punto {mathbf {q} }}}}sinistra(-{frac {parziale ^{2}varfi }{(matematica parziale {q} )^{2}}}{punto {mathbf {q} }}^{2}T+{frac {parziale ^{2}varfi }{partial varepsilon partial mathbf {q} }}{punto {mathbf {q} }}-{frac {partial varphi }{matematica parziale {q} }}{ddot {mathbf {q} }}Tright),dt.end{allineato}}} Notice that the Euler–Lagrange equations imply {stile di visualizzazione {inizio{allineato}{frac {d}{dt}}sinistra({frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {partial varphi }{matematica parziale {q} }}{punto {mathbf {q} }}Tright)&=left({frac {d}{dt}}{frac {partial L}{parziale {punto {mathbf {q} }}}}Giusto){frac {partial varphi }{matematica parziale {q} }}{punto {mathbf {q} }}T+{frac {partial L}{parziale {punto {mathbf {q} }}}}sinistra({frac {d}{dt}}{frac {partial varphi }{matematica parziale {q} }}Giusto){punto {mathbf {q} }}T+{frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {partial varphi }{matematica parziale {q} }}{ddot {mathbf {q} }},T\[6pt]&={frac {partial L}{matematica parziale {q} }}{frac {partial varphi }{matematica parziale {q} }}{punto {mathbf {q} }}T+{frac {partial L}{parziale {punto {mathbf {q} }}}}sinistra({frac {parziale ^{2}varfi }{(matematica parziale {q} )^{2}}}{punto {mathbf {q} }}Giusto){punto {mathbf {q} }}T+{frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {partial varphi }{matematica parziale {q} }}{ddot {mathbf {q} }},T.end{allineato}}} Substituting this into the previous equation, si ottiene {stile di visualizzazione {inizio{allineato}0={frac {dI'}{dvarepsilon }}[0]={}&L[mathbf {q} [t_{2}],{punto {mathbf {q} }}[t_{2}],t_{2}]T-L[mathbf {q} [t_{1}],{punto {mathbf {q} }}[t_{1}],t_{1}]T-{frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {partial varphi }{matematica parziale {q} }}{punto {mathbf {q} }}[t_{2}]T+{frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {partial varphi }{matematica parziale {q} }}{punto {mathbf {q} }}[t_{1}]T\[6pt]&{}+int _{t_{1}}^{t_{2}}{frac {partial L}{matematica parziale {q} }}{frac {partial varphi }{partial varepsilon }}+{frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {parziale ^{2}varfi }{partial varepsilon partial mathbf {q} }}{punto {mathbf {q} }},dt.end{allineato}}} Again using the Euler–Lagrange equations we get {stile di visualizzazione {frac {d}{dt}}sinistra({frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {partial varphi }{partial varepsilon }}Giusto)= sinistra({frac {d}{dt}}{frac {partial L}{parziale {punto {mathbf {q} }}}}Giusto){frac {partial varphi }{partial varepsilon }}+{frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {parziale ^{2}varfi }{partial varepsilon partial mathbf {q} }}{punto {mathbf {q} }}={frac {partial L}{matematica parziale {q} }}{frac {partial varphi }{partial varepsilon }}+{frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {parziale ^{2}varfi }{partial varepsilon partial mathbf {q} }}{punto {mathbf {q} }}.} Substituting this into the previous equation, si ottiene {stile di visualizzazione {inizio{allineato}0={}&L[mathbf {q} [t_{2}],{punto {mathbf {q} }}[t_{2}],t_{2}]T-L[mathbf {q} [t_{1}],{punto {mathbf {q} }}[t_{1}],t_{1}]T-{frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {partial varphi }{matematica parziale {q} }}{punto {mathbf {q} }}[t_{2}]T+{frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {partial varphi }{matematica parziale {q} }}{punto {mathbf {q} }}[t_{1}]T\[6pt]&{}+{frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {partial varphi }{partial varepsilon }}[t_{2}]-{frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {partial varphi }{partial varepsilon }}[t_{1}].fine{allineato}}} From which one can see that {stile di visualizzazione a sinistra({frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {partial varphi }{matematica parziale {q} }}{punto {mathbf {q} }}-Lright)T-{frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {partial varphi }{partial varepsilon }}} is a constant of the motion, cioè., it is a conserved quantity. Since φ[q, 0] = q, noi abbiamo {stile di visualizzazione {frac {partial varphi }{matematica parziale {q} }}=1} and so the conserved quantity simplifies to {stile di visualizzazione a sinistra({frac {partial L}{parziale {punto {mathbf {q} }}}}{punto {mathbf {q} }}-Lright)T-{frac {partial L}{parziale {punto {mathbf {q} }}}}{frac {partial varphi }{partial varepsilon }}.} To avoid excessive complication of the formulas, this derivation assumed that the flow does not change as time passes. The same result can be obtained in the more general case.

Field-theoretic derivation Noether's theorem may also be derived for tensor fields φA where the index A ranges over the various components of the various tensor fields. These field quantities are functions defined over a four-dimensional space whose points are labeled by coordinates xμ where the index μ ranges over time (μ = 0) and three spatial dimensions (μ = 1, 2, 3). These four coordinates are the independent variables; and the values of the fields at each event are the dependent variables. Under an infinitesimal transformation, the variation in the coordinates is written {stile di visualizzazione x^{in }rightarrow xi ^{in }=x^{in }+delta x^{in }} whereas the transformation of the field variables is expressed as {displaystyle varphi ^{UN}rightarrow alpha ^{UN}sinistra(xi ^{in }Giusto)=varphi ^{UN}sinistra(x^{in }Giusto)+delta varphi ^{UN}sinistra(x^{in }Giusto),.} By this definition, the field variations δφA result from two factors: intrinsic changes in the field themselves and changes in coordinates, since the transformed field αA depends on the transformed coordinates ξμ. To isolate the intrinsic changes, the field variation at a single point xμ may be defined {displaystyle alpha ^{UN}sinistra(x^{in }Giusto)=varphi ^{UN}sinistra(x^{in }Giusto)+{sbarra {delta }}varfi ^{UN}sinistra(x^{in }Giusto),.} If the coordinates are changed, the boundary of the region of space–time over which the Lagrangian is being integrated also changes; the original boundary and its transformed version are denoted as Ω and Ω’, rispettivamente.

Noether's theorem begins with the assumption that a specific transformation of the coordinates and field variables does not change the action, which is defined as the integral of the Lagrangian density over the given region of spacetime. Expressed mathematically, this assumption may be written as {displaystyle int _{Omega ^{primo }}Lleft(alfa ^{UN},{alfa ^{UN}}_{,non },xi ^{in }Giusto)d^{4}xi -int _{Omega }Lleft(varfi ^{UN},{varfi ^{UN}}_{,non },x^{in }Giusto)d^{4}x=0} where the comma subscript indicates a partial derivative with respect to the coordinate(S) that follows the comma, per esempio.

{stile di visualizzazione {varfi ^{UN}}_{,sigma }={frac {partial varphi ^{UN}}{partial x^{sigma }}},.} Since ξ is a dummy variable of integration, and since the change in the boundary Ω is infinitesimal by assumption, the two integrals may be combined using the four-dimensional version of the divergence theorem into the following form {displaystyle int _{Omega }sinistra{sinistra[Lleft(alfa ^{UN},{alfa ^{UN}}_{,non },x^{in }Giusto)-Lleft(varfi ^{UN},{varfi ^{UN}}_{,non },x^{in }Giusto)Giusto]+{frac {parziale }{partial x^{sigma }}}sinistra[Lleft(varfi ^{UN},{varfi ^{UN}}_{,non },x^{in }Giusto)delta x^{sigma }Giusto]Giusto}d^{4}x=0,.} The difference in Lagrangians can be written to first-order in the infinitesimal variations as {stile di visualizzazione a sinistra[Lleft(alfa ^{UN},{alfa ^{UN}}_{,non },x^{in }Giusto)-Lleft(varfi ^{UN},{varfi ^{UN}}_{,non },x^{in }Giusto)Giusto]={frac {partial L}{partial varphi ^{UN}}}{sbarra {delta }}varfi ^{UN}+{frac {partial L}{parziale {varfi ^{UN}}_{,sigma }}}{sbarra {delta }}{varfi ^{UN}}_{,sigma },.} Tuttavia, because the variations are defined at the same point as described above, the variation and the derivative can be done in reverse order; they commute {stile di visualizzazione {sbarra {delta }}{varfi ^{UN}}_{,sigma }={sbarra {delta }}{frac {partial varphi ^{UN}}{partial x^{sigma }}}={frac {parziale }{partial x^{sigma }}}sinistra({sbarra {delta }}varfi ^{UN}Giusto),.} Using the Euler–Lagrange field equations {stile di visualizzazione {frac {parziale }{partial x^{sigma }}}sinistra({frac {partial L}{parziale {varfi ^{UN}}_{,sigma }}}Giusto)={frac {partial L}{partial varphi ^{UN}}}} the difference in Lagrangians can be written neatly as {stile di visualizzazione {inizio{allineato}&left[Lleft(alfa ^{UN},{alfa ^{UN}}_{,non },x^{in }Giusto)-Lleft(varfi ^{UN},{varfi ^{UN}}_{,non },x^{in }Giusto)Giusto]\[4pt]={}&{frac {parziale }{partial x^{sigma }}}sinistra({frac {partial L}{parziale {varfi ^{UN}}_{,sigma }}}Giusto){sbarra {delta }}varfi ^{UN}+{frac {partial L}{parziale {varfi ^{UN}}_{,sigma }}}{sbarra {delta }}{varfi ^{UN}}_{,sigma }={frac {parziale }{partial x^{sigma }}}sinistra({frac {partial L}{parziale {varfi ^{UN}}_{,sigma }}}{sbarra {delta }}varfi ^{UN}Giusto).fine{allineato}}} così, the change in the action can be written as {displaystyle int _{Omega }{frac {parziale }{partial x^{sigma }}}sinistra{{frac {partial L}{parziale {varfi ^{UN}}_{,sigma }}}{sbarra {delta }}varfi ^{UN}+Lleft(varfi ^{UN},{varfi ^{UN}}_{,non },x^{in }Giusto)delta x^{sigma }Giusto}d^{4}x=0,.} Since this holds for any region Ω, the integrand must be zero {stile di visualizzazione {frac {parziale }{partial x^{sigma }}}sinistra{{frac {partial L}{parziale {varfi ^{UN}}_{,sigma }}}{sbarra {delta }}varfi ^{UN}+Lleft(varfi ^{UN},{varfi ^{UN}}_{,non },x^{in }Giusto)delta x^{sigma }Giusto}=0,.} For any combination of the various symmetry transformations, the perturbation can be written {stile di visualizzazione {inizio{allineato}delta x^{in }&=varepsilon X^{in }\delta varphi ^{UN}&=varepsilon Psi ^{UN}={sbarra {delta }}varfi ^{UN}+varepsilon {matematico {l}}_{X}varfi ^{UN}fine{allineato}}} dove {stile di visualizzazione {matematico {l}}_{X}varfi ^{UN}} is the Lie derivative of φA in the Xμ direction. When φA is a scalar or {stile di visualizzazione {X^{in }}_{,non }=0} , {stile di visualizzazione {matematico {l}}_{X}varfi ^{UN}={frac {partial varphi ^{UN}}{partial x^{in }}}X^{in },.} These equations imply that the field variation taken at one point equals {stile di visualizzazione {sbarra {delta }}varfi ^{UN}=varepsilon Psi ^{UN}-varepsilon {matematico {l}}_{X}varfi ^{UN},.} Differentiating the above divergence with respect to ε at ε = 0 and changing the sign yields the conservation law {stile di visualizzazione {frac {parziale }{partial x^{sigma }}}j^{sigma }=0} where the conserved current equals {displaystyle j^{sigma }= sinistra[{frac {partial L}{parziale {varfi ^{UN}}_{,sigma }}}{matematico {l}}_{X}varfi ^{UN}-l,X^{sigma }Giusto]-sinistra({frac {partial L}{parziale {varfi ^{UN}}_{,sigma }}}Giusto)Psi ^{UN},.} Manifold/fiber bundle derivation Suppose we have an n-dimensional oriented Riemannian manifold, M and a target manifold T. Permettere {stile di visualizzazione {matematico {C}}} be the configuration space of smooth functions from M to T. (Più generalmente, 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 {displaystyle mathbb {R} } , representing time and the target space is the cotangent bundle of space of generalized positions. 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. Per esempio, if there are m real-valued scalar fields, {displaystyle varphi _{1},ldot ,varfi _{m}} , then the target manifold is {displaystyle mathbb {R} ^{m}} . If the field is a real vector field, then the target manifold is isomorphic to {displaystyle mathbb {R} ^{3}} .

Now suppose there is a functional {stile di visualizzazione {matematico {S}}:{matematico {C}}rightarrow mathbb {R} ,} called the action. (It takes values into {displaystyle mathbb {R} } , piuttosto che {displaystyle mathbb {C} } ; this is for physical reasons, and is unimportant for this proof.) To get to the usual version of Noether's theorem, we need additional restrictions on the action. We assume {stile di visualizzazione {matematico {S}}[varfi ]} is the integral over M of a function {stile di visualizzazione {matematico {l}}(varfi ,parziale _{in }varfi ,X)} called the Lagrangian density, depending on φ, its derivative and the position. In altre parole, for φ in {stile di visualizzazione {matematico {C}}} {stile di visualizzazione {matematico {S}}[varfi ],=,int _{M}{matematico {l}}[varfi (X),parziale _{in }varfi (X),X],d^{n}X.} Suppose we are given boundary conditions, cioè., a specification of the value of φ at the boundary if M is compact, or some limit on φ as x approaches ∞. Then the subspace of {stile di visualizzazione {matematico {C}}} consisting of functions φ such that all functional derivatives of {stile di visualizzazione {matematico {S}}} at φ are zero, questo è: {stile di visualizzazione {frac {delta {matematico {S}}[varfi ]}{delta varphi (X)}}ca 0} and that φ satisfies the given boundary conditions, is the subspace of on shell solutions. (See principle of stationary action) Adesso, suppose we have an infinitesimal transformation on {stile di visualizzazione {matematico {C}}} , generated by a functional derivation, Q such that {displaystyle Qleft[int _{N}{matematico {l}},matematica {d} ^{n}xdestra]approx int _{partial N}f^{in }[varfi (X),partial varphi ,partial partial varphi ,ldot ],ds_{in }} for all compact submanifolds N or in other words, {stile di visualizzazione Q[{matematico {l}}(X)]approx partial _{in }f^{in }(X)} per tutti x, where we set {stile di visualizzazione {matematico {l}}(X)={matematico {l}}[varfi (X),parziale _{in }varfi (X),X].} If this holds on shell and off shell, we say Q generates an off-shell symmetry. If this only holds on shell, we say Q generates an on-shell symmetry. Quindi, we say Q is a generator of a one parameter symmetry Lie group.

Adesso, for any N, because of the Euler–Lagrange theorem, on shell (and only on-shell), noi abbiamo {stile di visualizzazione {inizio{allineato}Qleft[int _{N}{matematico {l}},matematica {d} ^{n}xdestra]&=int _{N}sinistra[{frac {parziale {matematico {l}}}{partial varphi }}-parziale _{in }{frac {parziale {matematico {l}}}{parziale (parziale _{in }varfi )}}Giusto]Q[varfi ],matematica {d} ^{n}x+int _{partial N}{frac {parziale {matematico {l}}}{parziale (parziale _{in }varfi )}}Q[varfi ],matematica {d} S_{in }\&approx int _{partial N}f^{in },matematica {d} S_{in }.fine{allineato}}} Since this is true for any N, noi abbiamo {displaystyle parziale _{in }sinistra[{frac {parziale {matematico {l}}}{parziale (parziale _{in }varfi )}}Q[varfi ]-f^{in }Giusto]ca 0.} But this is the continuity equation for the current {displaystyle J^{in }} definito da:[15] {displaystyle J^{in },=,{frac {parziale {matematico {l}}}{parziale (parziale _{in }varfi )}}Q[varfi ]-f^{in },} which is called the Noether current associated with the symmetry. 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 (fornito, Certo, if M is noncompact, the currents fall off sufficiently fast at infinity).

Comments Noether's theorem is an on shell theorem: it relies on use of the equations of motion—the classical path. It reflects the relation between the boundary conditions and the variational principle. Assuming no boundary terms in the action, Noether's theorem implies that {displaystyle int _{partial N}J^{in }ds_{in }ca 0.} The quantum analogs of Noether's theorem involving expectation values (per esempio., {textstyle leftlangle int d^{4}x~partial cdot {textbf {J}}rightrangle =0} ) probing off shell quantities as well are the Ward–Takahashi identities.

Generalization to Lie algebras Suppose we have two symmetry derivations Q1 and Q2. Quindi, [Q1, Q2] is also a symmetry derivation. Let's see this explicitly. Let's say {stile di visualizzazione Q_{1}[{matematico {l}}]approx partial _{in }f_{1}^{in }} e {stile di visualizzazione Q_{2}[{matematico {l}}]approx partial _{in }f_{2}^{in }} Quindi, {stile di visualizzazione [Q_{1},Q_{2}][{matematico {l}}]=Q_{1}[Q_{2}[{matematico {l}}]]-Q_{2}[Q_{1}[{matematico {l}}]]approx partial _{in }f_{12}^{in }} where f12 = Q1[f2μ] − Q2[f1μ]. Così, {stile di visualizzazione j_{12}^{in }= sinistra({frac {parziale }{parziale (parziale _{in }varfi )}}{matematico {l}}Giusto)(Q_{1}[Q_{2}[varfi ]]-Q_{2}[Q_{1}[varfi ]])-f_{12}^{in }.} This shows we can extend Noether's theorem to larger Lie algebras in a natural way.

Generalization of the proof This applies to any local symmetry derivation Q satisfying QS ≈ 0, 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. Quindi, because of the variational principle (which does not apply to the boundary, by the way), the derivation distribution q generated by q[e][Φ(X)] = ε(X)Q[Φ(X)] satisfies q[e][S] 0 for every ε, or more compactly, q(X)[S] 0 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 is related to the version given above, assume that the action is the spacetime integral of a Lagrangian that only depends on φ and its first derivatives. Anche, assume {stile di visualizzazione Q[{matematico {l}}]approx partial _{in }f^{in }} Quindi, {stile di visualizzazione {inizio{allineato}q[varepsilon ][{matematico {S}}]&=int q[varepsilon ][{matematico {l}}]d^{n}x\[6pt]&=int left{sinistra({frac {parziale }{partial varphi }}{matematico {l}}Giusto)varepsilon Q[varfi ]+sinistra[{frac {parziale }{parziale (parziale _{in }varfi )}}{matematico {l}}Giusto]parziale _{in }(varepsilon Q[varfi ])Giusto}d^{n}x\[6pt]&=int left{varepsilon Q[{matematico {l}}]+parziale _{in }varepsilon left[{frac {parziale }{partial left(parziale _{in }varphi right)}}{matematico {l}}Giusto]Q[varfi ]Giusto},d^{n}x\[6pt]&approx int varepsilon partial _{in }sinistra{f^{in }-sinistra[{frac {parziale }{parziale (parziale _{in }varfi )}}{matematico {l}}Giusto]Q[varfi ]Giusto},d^{n}xend{allineato}}} per tutti {displaystyle varepsilon } .

Più generalmente, if the Lagrangian depends on higher derivatives, poi {displaystyle parziale _{in }sinistra[f^{in }-sinistra[{frac {parziale }{parziale (parziale _{in }varfi )}}{matematico {l}}Giusto]Q[varfi ]-2sinistra[{frac {parziale }{parziale (parziale _{in }parziale _{non }varfi )}}{matematico {l}}Giusto]parziale _{non }Q[varfi ]+parziale _{non }sinistra[sinistra[{frac {parziale }{parziale (parziale _{in }parziale _{non }varfi )}}{matematico {l}}Giusto]Q[varfi ]Giusto]-,dotsm right]ca 0.} Esempi Esempio 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, è: {stile di visualizzazione {inizio{allineato}{matematico {S}}[X]&=int Lleft[X(t),{punto {X}}(t)Giusto],dt\&=int left({frac {m}{2}}somma _{io=1}^{3}{punto {X}}_{io}^{2}-V(X(t))Giusto),dt.end{allineato}}} The first term in the brackets is the kinetic energy of the particle, while the second is its potential energy. Consider the generator of time translations Q = d/dt. In altre parole, {stile di visualizzazione Q[X(t)]={punto {X}}(t)} . The coordinate x has an explicit dependence on time, whilst V does not; consequently: {stile di visualizzazione Q[l]={frac {d}{dt}}sinistra[{frac {m}{2}}somma _{io}{punto {X}}_{io}^{2}-V(X)Giusto]=msum _{io}{punto {X}}_{io}{ddot {X}}_{io}-somma _{io}{frac {partial V(X)}{parziale x_{io}}}{punto {X}}_{io}} so we can set {stile di visualizzazione L={frac {m}{2}}somma _{io}{punto {X}}_{io}^{2}-V(X).} Quindi, {stile di visualizzazione {inizio{allineato}j&=sum _{io=1}^{3}{frac {partial L}{parziale {punto {X}}_{io}}}Q[X_{io}]-L\&=msum _{io}{punto {X}}_{io}^{2}-sinistra[{frac {m}{2}}somma _{io}{punto {X}}_{io}^{2}-V(X)Giusto]\[3pt]&={frac {m}{2}}somma _{io}{punto {X}}_{io}^{2}+V(X).fine{allineato}}} The right hand side is the energy, and Noether's theorem states that {displaystyle dj/dt=0} (cioè. the principle of conservation of energy is a consequence of invariance under time translations).

Più generalmente, if the Lagrangian does not depend explicitly on time, the quantity {somma dello stile di visualizzazione _{io=1}^{3}{frac {partial L}{parziale {punto {X}}_{io}}}{punto {X_{io}}}-l} (called the Hamiltonian) is conserved.

Esempio 2: Conservation of center of momentum Still considering 1-dimensional time, permettere {stile di visualizzazione {inizio{allineato}{matematico {S}}sinistra[{vec {X}}Giusto]&=int {matematico {l}}sinistra[{vec {X}}(t),{punto {vec {X}}}(t)Giusto]dt\[3pt]&=int left[somma _{alpha =1}^{N}{frac {m_{alfa }}{2}}sinistra({punto {vec {X}}}_{alfa }Giusto)^{2}-somma _{alfa

Se vuoi conoscere altri articoli simili a Teorema di Noether puoi visitare la categoria Calcolo delle variazioni.

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