Document

19/06/2017
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
2n
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 inx 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
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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 BR
birTvis SigniT. SemoviRoT funqcia
w( x, )  E ( x, )  E ( y, ) , y    R .
cxadia, roca x  BR 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 gantolebis 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.
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