Translation examples
Matemaattisesti tämä oletus voidaan kirjoittaa seuraavasti: ∫ Ω ′ L ( α A , α A , ν , ξ μ ) d 4 ξ − ∫ Ω L ( ϕ A , ϕ A , ν , x μ ) d 4 x = 0 {\displaystyle \int _{\Omega ^{\prime }}L\left(\alpha ^{A},{\alpha ^{A}}_{,\nu },\xi ^{\mu }\right)d^{4}\xi -\int _{\Omega }L\left(\phi ^{A},{\phi ^{A}}_{,\nu },x^{\mu }\right)d^{4}x=0} missä muuttujien jälkeen yläpuolelle kirjoitetut pilkut tarkoittavat osittaisderivaattoja niiden koordinaattien suhteen, jotka seuraavat pilkun jälkeen, toisin sanoen ϕ A , σ = ∂ ϕ A ∂ x σ . {\displaystyle {\phi ^{A}}_{,\sigma }={\frac {\partial \phi ^{A}}{\partial x^{\sigma }}}\,.} Koska ξ on pelkkä integroimisvakio ja koska rajan Ω muutos oletettiin infinitesimaaliseksi, nämä kaksi integraalia voidaan yhdistää divergenssilauseen neliulotteisen version mukaisesti seuraavaan muotoon: ∫ Ω { + ∂ ∂ x σ } d 4 x = 0 . {\displaystyle \int _{\Omega }\left\{\left+{\frac {\partial }{\partial x^{\sigma }}}\left\right\}d^{4}x=0\,.} Lagrangen funktioiden erotus voidaan kirjoittaa ensimmäisessä kertaluvuissa infinitesimaalisilla muutoksilla: = ∂ L ∂ ϕ A δ ¯ ϕ A + ∂ L ∂ ϕ A , σ δ ¯ ϕ A , σ . {\displaystyle \left={\frac {\partial L}{\partial \phi ^{A}}}{\bar {\delta }}\phi ^{A}+{\frac {\partial L}{\partial {\phi ^{A}}_{,\sigma }}}{\bar {\delta }}{\phi ^{A}}_{,\sigma }\,.} Koska nämä muutokset kuitenkin on määritelty samassa edellä selityssä pisteessä, muutokset ja derivoinnit voidaan suorittaa myös päinvastaisessa järjestyksessä; ne kommutoivat: δ ¯ ϕ A , σ = δ ¯ ∂ ϕ A ∂ x σ = ∂ ∂ x σ ( δ ¯ ϕ A ) . {\displaystyle {\bar {\delta }}{\phi ^{A}}_{,\sigma }={\bar {\delta }}{\frac {\partial \phi ^{A}}{\partial x^{\sigma }}}={\frac {\partial }{\partial x^{\sigma }}}\left({\bar {\delta }}\phi ^{A}\right)\,.} Käyttämällä Eulerin-Lagrangen kenttäyhtälöä ∂ ∂ x σ ( ∂ L ∂ ϕ A , σ ) = ∂ L ∂ ϕ A {\displaystyle {\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\phi ^{A}}_{,\sigma }}}\right)={\frac {\partial L}{\partial \phi ^{A}}}} Lagrangen funktioiden erotus voidaan kirjoittaa yksinkertaisesti muotoon = ∂ ∂ x σ ( ∂ L ∂ ϕ A , σ ) δ ¯ ϕ A + ∂ L ∂ ϕ A , σ δ ¯ ϕ A , σ = ∂ ∂ x σ ( ∂ L ∂ ϕ A , σ δ ¯ ϕ A ) . {\displaystyle \left={\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\phi ^{A}}_{,\sigma }}}\right){\bar {\delta }}\phi ^{A}+{\frac {\partial L}{\partial {\phi ^{A}}_{,\sigma }}}{\bar {\delta }}{\phi ^{A}}_{,\sigma }={\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\phi ^{A}}_{,\sigma }}}{\bar {\delta }}\phi ^{A}\right)\,.} Näin ollen aktion muutokseksi saadaan ∫ Ω ∂ ∂ x σ { ∂ L ∂ ϕ A , σ δ ¯ ϕ A + L ( ϕ A , ϕ A , ν , x μ ) δ x σ } d 4 x = 0 . {\displaystyle \int _{\Omega }{\frac {\partial }{\partial x^{\sigma }}}\left\{{\frac {\partial L}{\partial {\phi ^{A}}_{,\sigma }}}{\bar {\delta }}\phi ^{A}+L\left(\phi ^{A},{\phi ^{A}}_{,\nu },x^{\mu }\right)\delta x^{\sigma }\right\}d^{4}x=0\,.} Koska tämä pätee missä tahansa alueessa Ω, integrandin on oltava nolla ∂ ∂ x σ { ∂ L ∂ ϕ A , σ δ ¯ ϕ A + L ( ϕ A , ϕ A , ν , x μ ) δ x σ } = 0 . {\displaystyle {\frac {\partial }{\partial x^{\sigma }}}\left\{{\frac {\partial L}{\partial {\phi ^{A}}_{,\sigma }}}{\bar {\delta }}\phi ^{A}+L\left(\phi ^{A},{\phi ^{A}}_{,\nu },x^{\mu }\right)\delta x^{\sigma }\right\}=0\,.} .
Expressed mathematically, this assumption may be written as ∫ Ω ′ L ( α A , α A , ν , ξ μ ) d 4 ξ − ∫ Ω L ( φ A , φ A , ν , x μ ) d 4 x = 0 {\displaystyle \int _{\Omega ^{\prime }}L\left(\alpha ^{A},{\alpha ^{A}}_{,\nu },\xi ^{\mu }\right)d^{4}\xi -\int _{\Omega }L\left(\varphi ^{A},{\varphi ^{A}}_{,\nu },x^{\mu }\right)d^{4}x=0} where the comma subscript indicates a partial derivative with respect to the coordinate(s) that follows the comma, e.g. φ A , σ = ∂ φ A ∂ x σ . {\displaystyle {\varphi ^{A}}_{,\sigma }={\frac {\partial \varphi ^{A}}{\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 ∫ Ω { + ∂ ∂ x σ } d 4 x = 0 . {\displaystyle \int _{\Omega }\left\{\left+{\frac {\partial }{\partial x^{\sigma }}}\left\right\}d^{4}x=0\,.} The difference in Lagrangians can be written to first-order in the infinitesimal variations as = ∂ L ∂ φ A δ ¯ φ A + ∂ L ∂ φ A , σ δ ¯ φ A , σ . {\displaystyle \left={\frac {\partial L}{\partial \varphi ^{A}}}{\bar {\delta }}\varphi ^{A}+{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}{\varphi ^{A}}_{,\sigma }\,.} However, 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 δ ¯ φ A , σ = δ ¯ ∂ φ A ∂ x σ = ∂ ∂ x σ ( δ ¯ φ A ) . {\displaystyle {\bar {\delta }}{\varphi ^{A}}_{,\sigma }={\bar {\delta }}{\frac {\partial \varphi ^{A}}{\partial x^{\sigma }}}={\frac {\partial }{\partial x^{\sigma }}}({\bar {\delta }}\varphi ^{A})\,.} Using the Euler–Lagrange field equations ∂ ∂ x σ ( ∂ L ∂ φ A , σ ) = ∂ L ∂ φ A {\displaystyle {\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}\right)={\frac {\partial L}{\partial \varphi ^{A}}}} the difference in Lagrangians can be written neatly as = ∂ ∂ x σ ( ∂ L ∂ φ A , σ ) δ ¯ φ A + ∂ L ∂ φ A , σ δ ¯ φ A , σ = ∂ ∂ x σ ( ∂ L ∂ φ A , σ δ ¯ φ A ) . {\displaystyle {\begin{aligned}&\left\\={}&{\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}\right){\bar {\delta }}\varphi ^{A}+{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}{\varphi ^{A}}_{,\sigma }={\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}\varphi ^{A}\right).\end{aligned}}} Thus, the change in the action can be written as ∫ Ω ∂ ∂ x σ { ∂ L ∂ φ A , σ δ ¯ φ A + L ( φ A , φ A , ν , x μ ) δ x σ } d 4 x = 0 . {\displaystyle \int _{\Omega }{\frac {\partial }{\partial x^{\sigma }}}\left\{{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}\varphi ^{A}+L\left(\varphi ^{A},{\varphi ^{A}}_{,\nu },x^{\mu }\right)\delta x^{\sigma }\right\}d^{4}x=0\,.} Since this holds for any region Ω, the integrand must be zero ∂ ∂ x σ { ∂ L ∂ φ A , σ δ ¯ φ A + L ( φ A , φ A , ν , x μ ) δ x σ } = 0 . {\displaystyle {\frac {\partial }{\partial x^{\sigma }}}\left\{{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}\varphi ^{A}+L\left(\varphi ^{A},{\varphi ^{A}}_{,\nu },x^{\mu }\right)\delta x^{\sigma }\right\}=0\,.} For any combination of the various symmetry transformations, the perturbation can be written δ x μ = ε X μ {\displaystyle \delta x^{\mu }=\varepsilon X^{\mu }} δ φ A = ε Ψ A = δ ¯ φ A + ε L X φ A {\displaystyle \delta \varphi ^{A}=\varepsilon \Psi ^{A}={\bar {\delta }}\varphi ^{A}+\varepsilon {\mathcal {L}}_{X}\varphi ^{A}} where L X φ A {\displaystyle {\mathcal {L}}_{X}\varphi ^{A}} is the Lie derivative of φA in the Xμ direction.
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