Simetriya u fiziciPeretvorennya Vidpovidnainvariantnist Vidpovidnijzakonzberezhennya Translyaciyi chasu Odnoridnistchasu energiyi C P CP i T simetriyi Izotropnistchasu parnosti Translyaciyi prostoru Odnoridnistprostoru impulsu Obertannya prostoru Izotropnistprostoru momentuimpulsu Grupa Lorenca busti VidnosnistLorenc kovariantnist ruhucentra mas Kalibruvalneperetvorennya Kalibruvalnainvariantnist zaryaduTeorema Neter tverdzhennya v teoretichnij fizici zgidno z yakim kozhnij diferencijovnij simetriyi vidpovidaye integral ruhu Napriklad odnoridnosti prostoru vidpovidaye zakon zberezhennya impulsu Odnoridnist prostoru oznachaye te sho pri perenesenni fizichnoyi sistemi na bud yakij vektor v bud yakomu napryamku vsi fizichni procesi v nij ne zminyatsya Vidpovidno inshi tipi simetriyi mayut svoyi integrali ruhu odnoridnist chasu zakon zberezhennya energiyi izotropnist prostoru zakon zberezhennya momentu impulsu kalibruvalna invariantnist zakon zberezhennya elektrichnogo zaryadu Teoremu sformulyuvala j dovela 1918 roku nimecka matematikinya Emmi Neter Dovedennya teoremi Neter RedaguvatiDovedennya teoremi Neter Nehaj u prostori chasi v yakomu zapisanij viraz dlya diyi isnuyut pevni simetriyi Ce oznachaye sho integral cherez yakij viznachayetsya diya ne zminyuyetsya pri zastosuvanni deyakih neperervnih peretvoren kozhne z yakih vidpovidaye svoyij simetriyi Ce oznachaye sho ne zminyuyutsya i rivnyannya ruhu Shukani peretvorennya yaki zadovolnuyut cij umovi i treba znajti Nehaj ye deyaki neperervni peretvorennya koordinat x m displaystyle x mu nbsp ta poliv PS m displaystyle Psi mu nbsp sho zalezhat vid m displaystyle m nbsp dijsnih parametriv w a w 1 w m displaystyle omega alpha omega 1 omega m nbsp Todi x m x m f m x w a PS k x PS k x F k PS x w a 1 displaystyle x mu x mu f mu x omega alpha quad Psi k x Psi k x F k Psi x omega alpha qquad 1 nbsp prichomu pri totozhnih peretvorennyah mozhna zapisati sho f x m 0 x m F PS x 0 PS x 2 displaystyle f x mu 0 x mu F Psi x 0 Psi x qquad 2 nbsp a umovoyu invariantnosti diyi pri peretvorennyah cih velichin ye w L PS x m PS x d 4 x w L PS x m PS x d 4 x 3 displaystyle int limits omega L Psi x partial mu Psi x d 4 x int limits omega L Psi x partial mu Psi x d 4 x qquad 3 nbsp de vrahovana zalezhnist funkciyi Lagranzha vid yak vid chasu tak i vid tochki u chastinnomu vipadku koli funkciya Lagranzha zapisana dlya skalyarnih funkcij vona zalezhit lishe vid chasu prote u bilsh zagalnomu vipadku vona zapisana dlya vektornih funkcij a otzhe zalezhit vid 4 vektora r m c t r displaystyle r mu ct mathbf r nbsp Dlya malih w a displaystyle omega alpha nbsp z urahuvannyam 2 displaystyle 2 nbsp 1 displaystyle 1 nbsp mozhna rozklasti v ryad do linijnih po w a displaystyle omega alpha nbsp dodankiv f x m w a x a f x m w a w a w a x m d x m F k PS k x w a k PS k x a k PS k w a w a k PS k x k d PS k x 4 displaystyle f x mu omega alpha approx x sum limits alpha frac partial f x mu omega alpha partial omega alpha omega alpha x mu delta x mu quad F k Psi k x w alpha approx sum limits k Psi k x sum limits alpha k frac partial Psi k partial omega alpha omega alpha sum limits k Psi k x sum limits k delta Psi k x qquad 4 nbsp U podalshomu poznachennya sumi dlya cih i pov yazanih iz nimi viraziv ne budut pisatisya Dlya togo shob vidiliti u peretvorenni polya peretvorennya sho zminyuye funkcionalnu zalezhnist polya vid argumentiv i peretvorennya sho zminyuye znachennya poliv mozhna rozklasti pole dlya malogo prirostu koordinati d x displaystyle delta x nbsp PS k x PS k x d x PS k x d x m m PS k x 5 displaystyle Psi k x approx Psi k x delta x Psi k x delta x mu partial mu Psi k x qquad 5 nbsp de shtrih pri PS k displaystyle Psi k nbsp u drugomu dodanku pribrano dlya zberezhennya poryadku malosti po w a displaystyle omega alpha nbsp Z inshogo boku sravedlivij viraz 4 displaystyle 4 nbsp Yaksho pririvnyati 4 displaystyle 4 nbsp do 5 displaystyle 5 nbsp mozhna otrimati PS k x d x m m PS k x PS k x d PS k x PS k x PS k x d PS k x d x m m PS k x PS k x d PS k x 6 displaystyle Psi k x delta x mu partial mu Psi k x Psi k x delta Psi k x Rightarrow Psi k x Psi k x delta Psi k x delta x mu partial mu Psi k x Psi k x tilde delta Psi k x qquad 6 nbsp Iz poznachennya PS x displaystyle Psi x nbsp vidno sho 6 displaystyle 6 nbsp vidpovidaye za zminu formi funkciyi bez zmini funkcionalnoyi zalezhnosti vid x displaystyle x nbsp Koristuyuchis zaminoyu zminnih pri integruvanni mozhna otrimati sho w L PS x m PS x d 4 x w L PS x m PS x m L PS x m PS x d x m d 4 x 7 displaystyle int limits omega L Psi x partial mu Psi x d 4 x approx int limits omega left L Psi x partial mu Psi x partial mu L Psi x partial mu Psi x delta x mu right d 4 x qquad 7 nbsp Dovedennya Danij rozklad stosuyetsya lishe rozkladu po zmini funkcionalnoyi zalezhnosti polya a ne zmini formi samogo polya tomu chisto formalno u ramkah dovedennya mozhna zrobiti perepoznachennya L PS x m PS x gt G x displaystyle L Psi x partial mu Psi x gt G x nbsp Todi w G x d 4 x w G x d x I d 4 x displaystyle int limits omega G x d 4 x int limits omega G x delta x Id 4 x nbsp Dlya neskinchenno malogo peretvorennya Yakobian I x n x m displaystyle I frac partial x nu partial x mu nbsp zi zberezhennyam linijnosti po w a displaystyle omega alpha nbsp riven rozglyadayetsya vipadok dvovimirnogo prostoru chasu prote usi nastupni peretvorennya spravedlivi i dlya chotirivimirnogo u chomu mozhna vdostoviritis pri bezposerednij perevirci I x n d x n x m 0 x 0 d x 0 0 x 1 d x 1 1 x 0 d x 0 1 x 1 d x 1 1 0 d x 0 0 d x 1 1 d x 0 1 1 d x 1 1 m d x m displaystyle I frac partial x nu delta x nu partial x mu begin vmatrix partial 0 x 0 delta x 0 amp partial 0 x 1 delta x 1 partial 1 x 0 delta x 0 amp partial 1 x 1 delta x 1 end vmatrix begin vmatrix 1 partial 0 delta x 0 amp partial 0 delta x 1 partial 1 delta x 0 amp 1 partial 1 delta x 1 end vmatrix approx 1 partial mu delta x mu nbsp Todi z urahuvannyam 5 displaystyle 5 nbsp w G x d x I d 4 x w G x d x m G x 1 m d x m d 4 x w G x G x m d x m d x m G x d 4 x w G x m d x G x d 4 x displaystyle int limits omega G x delta x Id 4 x int limits omega G x delta x partial mu G x 1 partial mu delta x mu d 4 x approx int limits omega G x G x partial mu delta x mu delta x partial mu G x d 4 x int limits omega G x partial mu delta xG x d 4 x nbsp Povertayuchis do zaminenoyi funkciyi mozhna otrimati w L PS x m PS x d 4 x w L PS x m PS x m L PS x m PS x d x m d 4 x displaystyle int limits omega L Psi x partial mu Psi x d 4 x approx int limits omega left L Psi x partial mu Psi x partial mu L Psi x partial mu Psi x delta x mu right d 4 x nbsp de shtrih pri PS displaystyle Psi nbsp dlya drugogo dodanku z L PS x m PS x displaystyle L Psi x partial mu Psi x nbsp pribrano dlya zberezhennya pershogo poryadku malosti Zalishayetsya lishe peretvoriti dodanok L PS x PS x displaystyle L Psi x partial Psi x nbsp L PS x m PS x 6 L PS x d PS x PS x d PS x L PS x m PS x L PS k d PS k L m PS k x m d PS k x displaystyle L Psi x partial mu Psi x 6 L Psi x tilde delta Psi x partial Psi x partial tilde delta Psi x approx L Psi x partial mu Psi x frac partial L partial Psi k tilde delta Psi k frac partial L partial partial mu Psi k x partial mu tilde delta Psi k x nbsp Pidstavivshi ce u 7 displaystyle 7 nbsp mozhna otrimati w L PS x PS x L PS k d PS k L m PS k x m d PS k x m L PS x m PS x d x m d 4 x 3 w L PS x m PS x d 4 x displaystyle int limits omega left L Psi x partial Psi x frac partial L partial Psi k tilde delta Psi k frac partial L partial partial mu Psi k x partial mu tilde delta Psi k x partial mu L Psi x partial mu Psi x delta x mu right d 4 x 3 int limits omega L Psi x partial mu Psi x d 4 x Rightarrow nbsp w L PS k d PS k x L m PS k x m d PS k x m L PS x m PS x d x m d 4 x 0 8 displaystyle Rightarrow int limits omega left frac partial L partial Psi k tilde delta Psi k x frac partial L partial partial mu Psi k x partial mu tilde delta Psi k x partial mu L Psi x partial mu Psi x delta x mu right d 4 x 0 qquad 8 nbsp Vikoristovuyuchi rivnyannya Lagranzha dlya vektornih poliv m L m PS k L PS k displaystyle partial mu left frac partial L partial partial mu Psi k right frac partial L partial Psi k nbsp iz 8 displaystyle 8 nbsp mozhna otrimati w m L m PS k d PS k x L m PS k x m d PS k x m L PS x m PS x d x m d 4 x 0 9 displaystyle int limits omega left partial mu left frac partial L partial partial mu Psi k right tilde delta Psi k x frac partial L partial partial mu Psi k x partial mu tilde delta Psi k x partial mu L Psi x partial mu Psi x delta x mu right d 4 x 0 qquad 9 nbsp Teper mozhna vinesti m displaystyle partial mu nbsp iz virazu u duzhkah oskilki iz 6 displaystyle 6 nbsp sliduye sho d PS k PS k x PS k x d m PS k m PS k m PS k m d PS k m L m PS k d PS k x L m PS k x m d PS k x displaystyle tilde delta Psi k Psi k x Psi k x Rightarrow tilde delta partial mu Psi k partial mu Psi k partial mu Psi k partial mu tilde delta Psi k Rightarrow partial mu left frac partial L partial partial mu Psi k right tilde delta Psi k x frac partial L partial partial mu Psi k x partial mu tilde delta Psi k x nbsp m L m PS k d PS k x displaystyle partial mu left frac partial L partial partial mu Psi k tilde delta Psi k x right nbsp Otzhe w m L m PS k d PS k x m L PS x m PS x d x m d 4 x d PS k PS k x PS k x d x n X a n w a d PS k x Y k a w a displaystyle int limits omega partial mu left frac partial L partial partial mu Psi k tilde delta Psi k x partial mu L Psi x partial mu Psi x delta x mu right d 4 x tilde delta Psi k Psi k x Psi k x quad delta x nu X alpha nu omega alpha quad delta Psi k x Y k alpha omega alpha nbsp w m L m PS k Y k a X a n n PS k L X a n w a d 4 x 0 9 displaystyle int limits omega partial mu left frac partial L partial partial mu Psi k left Y k alpha X alpha nu partial nu Psi k right LX alpha nu right omega alpha d 4 x 0 qquad 9 nbsp Otzhe yaksho diya invariantna vidnosno deyakih peretvoren koordinat ta poliv x f x w a PS x F PS x w a displaystyle x f x omega alpha quad Psi x F Psi x omega alpha nbsp a 1 m displaystyle alpha 1 m nbsp to isnuye m velichin J a m displaystyle J alpha mu nbsp J a m L m PS k Y k a L m PS k n PS k d n m L X a n displaystyle J alpha mu frac partial L partial partial mu Psi k Y k alpha left frac partial L partial partial mu Psi k partial nu Psi k delta nu mu L right X alpha nu nbsp d n m displaystyle delta nu mu nbsp simvol Kronekera prichomu z 9 displaystyle 9 nbsp vidno sho m J a m 0 J a 0 j J a j J a 0 t j J a j 0 10 displaystyle partial mu J alpha mu partial 0 J alpha 0 partial j J alpha j frac partial J alpha 0 partial t nabla j J alpha j 0 qquad 10 nbsp Otzhe mozhna provesti analogiyu J a m displaystyle J alpha mu nbsp zi strumami oskilki ostannye rivnyannya ye rivnyannyam neperervnosti Z nogo zh mozhna otrimati sho J a 0 t d Q a d t j J a j J a j d S Q a V J a 0 t r d 3 r 11 displaystyle frac partial J alpha 0 partial t frac dQ alpha dt int nabla j J alpha j int J alpha j dS Q alpha int limits V J alpha 0 t mathbf r d 3 mathbf r qquad 11 nbsp de velichina Q displaystyle Q nbsp umovno nazvana zaryadom a prostorovi komponenti J a j displaystyle J alpha j nbsp mozhna interpretuvati yak deyakij vektor potoku sho zminyuye ci zaryadi pri peresikanni potokom poverhni S displaystyle S nbsp sho obmezhuye ob yem V displaystyle V nbsp Yaksho polya na neskinchennosti znikayut to z 11 displaystyle 11 nbsp sliduye sho d Q a d t 0 displaystyle frac dQ alpha dt 0 nbsp Oskilki pri vidnesenni poverhni S displaystyle S nbsp na neskinchennist sliduye sho potik J a j displaystyle J alpha j nbsp cherez neyi riven nulyu Dinamichni invarianti RedaguvatiTenzor energiyi impulsu Tenzor orbitalnogo momentu Tenzor spinovogo momentu Z pershogo tenzoru mozhna otrimati dinamichnij invariant yakij nazivayetsya 4 vektor energiyi impulsu Z drugogo i tretogo tenzoru otrimuyut psevdovektori orbitalnogo momentu i spinu vidpovidno Pri comu vikoristovuyut zgortku z absolyutno antisimetrichnim tenzorom Levi Chiviti Div takozh RedaguvatiZakoni zberezhennya Tenzor energiyi impulsu Spryazheni zminni nbsp Ce nezavershena stattya z fiziki Vi mozhete dopomogti proyektu vipravivshi abo dopisavshi yiyi Cya stattya ne mistit posilan na dzherela Vi mozhete dopomogti polipshiti cyu stattyu dodavshi posilannya na nadijni avtoritetni dzherela Material bez dzherel mozhe buti piddano sumnivu ta vilucheno sichen 2017 Otrimano z https uk wikipedia org w index php title Teorema Neter amp oldid 38785242
