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27/06/2017 maTematikuri fizikis gantolebebi leqcia 7 3.3. potencialTa Teoria 3.3.1. potencialebi. vTqvaT, R n SemosazRvruli area A k ,h , k 1 , klasis sazRvriT da u C 2 . maSin, rogorc vaCveneT, nebismier wertilSi u funqciis mniSvneloba ganisazRvreba E ( x, ) u ( x) (3.3.1) u ( ) E ( x, )u ( x)dx u ( x) E ( x, ) dS x formuliT, sadac E ( x, ) laplasis gantolebis fundamenturi amonaxsnia polusiT wertilSi. Tu u f ( x ) , (3.3.2) maSin (3.3.2)-is (3.3.1)-Si CasmiT miviRebT, rom E ( x, ) u ( x) u ( ) E ( x, ) f ( x)dx u ( x) E ( x, ) dS x . (3.3.3) am formulaSi Semaval E ( x, ) dS x da V ( x) ( x) E ( x, )dS x U ( x) f ( x) E ( x, )dx , W ( x) ( x) integralebs Sesabamisad moculobiTi (niutonis), ormagi fenis da martivi fenis potenciali ewodeba. f ( x ) da ( x ) funqciebs potencialebis simkvrive ewodeba. 3.3.2. moculobiTi potenciali. ganvixiloT U ( x) f ( ) E ( x, )d moculobiTi potenciali f simkvriviT. radgan E ( x, ) harmoniulia, roca x , amitom U 2n moculobiTi potenciali harmoniulia R n \ areSi. amasTan, radgan E ( x , ) ~ x , roca x , advili dasamtkicebelia, rom n 3 SemTxvevaSi u( x) , roca x . Teorema 3.3.1. Tu f funqcia uwyveti da SemosazRvrulia areSi, maSin U moculobiTi potenciali uwyveti da uwyvetad warmoebadia R n -Si. amasTan E (3.3.4) U , j ( x) f ( ) ( x, )d , j 1,2,, n . x j E 1 n SevniSnoT, rom radgan ( x , ) ~ x , roca x , amitom es arasakuTrivi inx j tegrali Tanabrad krebadia. Teorema 3.3.2. Tu f simkvrives areSi SemosazRvruli da uwyveti pirveli rigis warmoebulebi aqvs, maSin moculobiTi potencials aqvs uwyveti meore rigis warmoebulebi areSi da E E U , j j ( x) f ( ) ( x, ) cos , j dS f , j ( ) ( x, )d , (3.3.5) j sadac aris zedapirisadmi wertilSi gavlebuli gare normali. damtkiceba. cxadia, E E ( x, ) ( x, ) , xj j amitom (3.3.4) formulidan miiReba, rom j leqcia 7 maTematikuri fizikis gantolebebi U , j ( x) f ( ) E ( x, )d xj nawilobiTi integrebis Semdeg miviRebT, rom U , j ( x) f ( ) E ( x, ) cos , j dS f , j ( ) E ( x, )d . (3.3.6) advili dasanaxia, rom roca x , maSin (3.3.6) formulis marjvena mxaris pirveli integrali warmoebadia x j -s mimarT, rogorc parametrze damokidebuli sakuTrivi integrali da xj E f ( ) E ( x, ) cos , j dS f ( ) ( x, ) cos , j dS . x j rac Seexeba marjvena mxareSi mdgom meore integrals, esaa moculobiTi potenciali, romlis simkvrive f , j ( ) SemosazRvruli da uwyveti funqciaa areSi. amitom, wina Teoremis Tanaxmad, xj f ,j ( ) E ( x, )d f , j ( ) E ( x, )d . xj amrigad, U funqciis meore rigis warmoebulebi uwyvetia areSi da gamoiTvleba Semdegi formuliT E E U , j j ( x) f ( ) ( x, ) cos , j dS f, j ( ) ( x, )d xj xj f ( ) E E ( x, ) cos , j dS f, j ( ) ( x, )d . j j 3.3.3. puasonis gantolebis amoxsna. Tu (3.3.5) formulaSi avjamavT j-s mimarT 1-dan nmde, gveqneba U ( x) f ( ) E E ( x, ) cos , j dS f , j ( ) ( x, )d . j j aqedan gamomdinareobs, rom U ( x) f ( ) E E ( x, )dS lim f , j ( ) ( x, )d , 0 j (3.3.7) sadac \ B x . advili dasanaxia, rom n E E 2E E f f , j ( ) ( x, ) f ( ) 2 ( x, ) f , j ( ) , j j j j j j 1 radgan areSi E 0 , amitom (3.3.7) formulidan miviRebT, rom E E U ( x) f ( ) ( x, )dS lim f d . 0 j j bolo integralSi gaus-ostrogradskis formulis gamoyenebiT da (3.1.9) formulis miRebis dros gamoyenebuli msjelobiT gveqneba, rom E E E U ( x) f ( ) ( x, )dS lim f ( ) ( x, )dS f ( ) ( x, )dS 0 S x 2 leqcia 7 maTematikuri fizikis gantolebebi lim 0 E f ( ) ( x, )dS f ( x) . S x amrigad, U ( x ) f ( x ) , (3.3.8) riTac davamtkiceT Semdegi Teorema 3.3.3. Teorema 3.3.2-is pirobebSi adgili aqvs (3.3.8) tolobas, e. i. moculobiTi potenciali akmayofilebs puasonis (3.3.8) gantolebas. Teorema 3.3.4. Tu G( x, ) funqcia aris areSi laplasis operatorisTvis dirixles amocanis grinis funqcia daìf SemosazRvruli funqciaa, romelsac areSi gaaCnia SemosazRvruli da uwyveti pirveli rigis warmoebulebi, maSin (3.3.9) u( x) f ( )G( x, )d tolobiT gansazRvruliìu funqcia A , k 1 , klasis areSi warmoadgens puasonis (3.3.8) gantolebis amonaxsns, romelic akmayofilebs lim0 u( x) 0 (3.3.10) k ,h x x sasazRvro pirobas. damtkiceba. radgan G( x, ) E ( x, ) g( x, ) , sadac g harmoniulia areSi da G( x, ) 0 , Teorema 3.3.3-is Tanaxmad, U ( x) f ( ) E ( x, )d f ( ) g ( x, )d f ( ) E ( x, )d f ( ) x . 0 marTalia, G( x, ) 0 , roca x x , magram (3.3.9) formulaSi pirdapir zRvarze ver gadavalT, radgan ar viciT, aris Tu ara es krebadoba Tanabari. amitom u funqcia warmovadginoT u ( x) f ( )G ( x, )d f ( )G ( x, )d saxiT, sadac d d : x 0 da \ d (ix. nax. 3.3.1). radgan pirveli integrali sakuTrivi integralia, roca x x 0 x 0 da , amasTan argumentebis imave mniSvnelobebisTvis integralqveSa gamosaxuleba uwyvetia ( x, ) -is mimarT, amitom pirvel integralSi zRvarze gadasvla integralis niSnis qveS SeiZleba: lim x x0 f ( )G( x, )d f ( ) lim G( x, )d 0 . x x0 3 leqcia 7 maTematikuri fizikis gantolebebi x0 d nax. 3.3.1 vaCvenoT, rom meore integralic nuliskenaa krebadi. aviRoT S R sfero imdenad didi radiusiT, rom nebismieri -sTvis moxvdes BR birTvis SigniT. SemoviRoT funqcia w( x, ) E ( x, ) E ( y, ) , y R . cxadia, roca x BR birTvs, w( x, ) 0 , xolo roca x -s, G( x, ) w( x, ) x w( x, ) x 0 . magram areSi G( x, ) w( x, ) funqcia harmoniulia. maSin maqsimumis principidan gamomdinareobs, rom G( x, ) w( x, ) 0 w( x, ) G( x, ) 0 , radgan G( x, ) 0 . amdenad, 0 G( x, )d w( x, )d , roca x d . d d magram ramdenadac w( x, ) funqciis integrali d areze Tanabrad krebadia, amitom w( x, )d 0 . d e. i., 0 G( x, )d 0 , d 0 saidanac f ( ) -s SemosazRvrulobis gamo f ( ) G( x, )d 0 . d 0 4 leqcia 7 maTematikuri fizikis gantolebebi maSasadame, meore integralic nulisken aris krebadi da amitom U ( x) 0 , roca x x 0 . rogorc am Teoremidan Cans, Tu grinis funqcia cnobilia, maSin moculobiTi potenciali gvaZlevs dirixles erTgvarovani amocanis amonaxsns puasonis gantolebisTvis. Tu sasazRvro piroba erTgvarovani araa, lim u( x ) x 0 , 0 x x maSin unda avarCioT iseTi v C 2 () funqcia, romelic sasazRvro pirobas daakmayofilebs. amis mere ganvixiloT w( x) u( x) v( x) funqcia, romelic daakmayofilebs erTgvarovan sasazRvro pirobas da w( x ) f ( x ) v ( x ) gantolebas. amdenad, mivediT ukve cxadad amoxsnil amocanamde. 5