1
�03=8B=K5 ?0@04>:AK 8 8E >1JOA=5=85.
.. >@=520, .. C;838=, .. C;838=0
AA;54>20B5;LA:0O 3@C??0  (http://kuligin.mylivepage.ru)
==>B0F8O.  AB0BL5 ?@82>48BAO 0=0;87 ?0@04>:A>2 :2078AB0B8G5A:>9 M;5:B@>48=0<8:8, 87;>65==KE 2
@01>B5 . 8:>;0520 �!>2@5<5==0O M;5:B@>48=0<8:0 8 ?@8G8=K 55 ?0@04>:A0;L=>AB8. >:070=>, GB>
1>;LH8=AB2> 87 =8E <>6=> A CA?5E>< >1JOA=8BL, >?8@0OAL =0 =LNB>=>2A:CN <5E0=8:C.
2545=85
5402=> =0< 2AB@5B8;0AL AB0BLO 5==048O 0A8;L528G0 8:>;0520 �!>2@5<5==0O
M;5:B@>48=0<8:0 8 ?@8G8=K 55 ?0@04>:A0;L=>AB8. (-:A?5@8<5=B0;L=K5 ?0@04>:AK
M;5:B@>48=0<8:8) [1].  AB0BL5 >?8A0=> 40 M:A?5@8<5=B0;L=KE @57C;LB0B>2, >1JOA=5=8O
:>B>@K<, :0: >= ?8H5B, 4> A8E ?>@ =5 =0945=> A>2@5<5==>9 =0C:>9.
 =59 02B>@ ?8H5B:
�N10O B5>@8O, :0:>9 1K A>25@H5==>9 >=0 =8 1K;0, =8:>340 =5 8A:;NG05B 2>7<>6=>AB8
40;L=59H53> A2>53> A>25@H5=AB2>20=8O. &5==>ABL 65 D878G5A:>9 B5>@88 >?@545;O5BAO,
?@5645 2A53>, >1;0ABLN 55 ?@0:B8G5A:>9 ?@8;>68<>AB8. @8<5=8B5;L=> : 8725AB=>9
B5>@88 M;5:B@><03=5B87<0, 2 @0<:0E A>2@5<5==KE ?@54AB02;5=89 2 M;5:B@>48=0<8:5,
2>?@>A > H8@>:>9 ?@0:B8G5A:>9 ?@8;>68<>AB8 B5>@88, :>=5G=>, =5 ?>4;568B A><=5=8N.
4=0:>, =5A<>B@O =0, :070;>AL 1K, 1573@0=8G=K5 >1;0AB8 ?@0:B8G5A:>9 ?@8;>68<>AB8
A>2@5<5==>9 B5>@88 M;5:B@><03=5B87<0 8 3@0=48>7=K5 4>AB865=8O =0C:8 8 B5E=8:8 2
MB8E >1;0ABOE, : =0AB>OI5<C 2@5<5=8 2 M;5:B@>48=0<8:5 =0:>?8;>AL 7=0G8B5;L=>5
:>;8G5AB2> O2;5=89 M;5:B@><03=5B87<0, :>B>@K< A>2@5<5==0O B5>@8O C65 =5 <>65B
40BL =5?@>B82>@5G82>3> 8 :>@@5:B=>3> >1JOA=5=8O. "> 5ABL 2 4;8B5;L=>< ?@>F5AA5
?@0:B8G5A:>3> >A2>5=8O B5E 70:>=>2 M;5:B@><03=5B87<0, :>B>@K5 1K;8 >?@545;5=K 2
@0<:0E 8725AB=>9 B5>@88, G5;>25G5AB2> AB0;:820;>AL 8 A B5<8 O2;5=8O<8
M;5:B@><03=5B87<0, :>B>@K5 C65 O2=> 2KE>48;8 70 @0<:8 A>2@5<5==>9 B5>@88. >O2;5=85
B0:8E ?0@04>:A0;L=KE A B>G:8 7@5=8O ACI5AB2CNI59 B5>@88 O2;5=89 M;5:B@><03=5B87<0
O2;O5BAO 2?>;=5 5AB5AB25==K< 8 :0: @07 4>:07K205B A>1>9, A >4=>9 AB>@>=K,
>3@0=8G5==>ABL ACI5AB2CNI59 B5>@88 M;5:B@><03=5B87<0, 0 A 4@C3>9 AB>@>=K,
=5>1E>48<>ABL 40;L=59H53> 55 A>25@H5=AB2>20=8O. >MB><C, GB>1K @07>1@0BLAO 2
?@8G8=0E ?0@04>:A0;L=>AB8 A>2@5<5==>9 M;5:B@>48=0<8:8, A;54C5B >1@0B8BLAO, ?@5645
2A53>, : 0=0;87C ACI5AB20 B5E =570A;C65==> 701KBKE ?0@04>:A0;L=KE O2;5=89
M;5:B@><03=5B87<0, :>B>@K5 1K;8 8725AB=K 5I5 2> 2@5<5=0 <?5@0 8 >1=0@C6820NBAO 2
=0AB>OI55 2@5<O 2 <=>3>G8A;5==KE M:A?5@8<5=B0;L=KE =01;N45=8OE. >;8G5AB2> B0:8E
4>?>;=ONI8E 4@C3 4@C30 =01;N45=89 =0:>?8;>AL C65 2?>;=5 4>AB0B>G=>, GB>1K <>6=>
1K;> A45;0BL ?> =8< 2?>;=5 >4=>7=0G=K5 2K2>4K.  4>?>;=5=85 : 8725AB=K< =01;N45=8O<
�AB@0==KE <03=8B=KE 2708<>459AB289 02B>@>< ?>AB02;5=0 A5@8O A?5F80;L=KE
M:A?5@8<5=B>2, :>B>@K<8 @0A:@K205BAO 459AB28B5;L=0O D878G5A:0O ACI=>ABL B0:
=07K205<KE �?0@04>:A0;L=KE O2;5=89 M;5:B@><03=5B87<0. 865 405BAO >?8A0=85
>1=0@C68205<KE 2 <=>3>G8A;5==KE M:A?5@8<5=B0E �AB@0==KE <03=8B=KE A8;
2708<>459AB28O, :>@@5:B=>3> >1JOA=5=8O :>B>@K< 2 @0<:0E A>2@5<5==>9 B5>@88
M;5:B@><03=5B87<0 =09B8 =5 C405BAO.
2B>@ ?@02 2 B><, GB> A>2@5<5==>5 A>AB>O=85 B5>@88 M;5:B@8G5AB20 (>D8F80;L=0O =0C:0)
459AB28B5;L=> =5 ?>72>;O5B 40BL ?@85<;5<>3> =5?@>B82>@5G82>3> >1JOA=5=8O. 865 <K
4048< >1JOA=5=85 =5:>B>@K< M:A?5@8<5=B0<, >?8@0OAL =0 0;LB5@=0B82=K5 ?@54AB02;5=8O.
 A8;C AE>65AB8 (?>4>18O) >1JOA=5=8O <K @0718;8 M:A?5@8<5=BK =0 =5A:>;L:> 3@C??.
'0AB8 M:A?5@8<5=B>2 <K =5 405< >1JOA=5=8O 87-70 >BACBAB28O ?>;=>BK 8E >?8A0=8O.
2
1. "5>@5B8G5A:85 ?@54?>AK;:8 4;O >1JOA=5=8O
0=55 ?@>2545==K5 =0<8 B5>@5B8G5A:85 8AA;54>20=8O ?>72>;ONB 40BL >1JOA=5=8O
?@82545==K< 2 [1] M:A?5@8<5=B0<. BAK;0O 70 ?>4@>1=>ABO<8 4>:070B5;LAB2 : @01>B0< [2],
[3], [4] 8 4@., <K ?@82545< :@0B:85 A2545=8O 87 MB8E @01>B.
1. 0: 1K;> ?>:070=> 2 @01>B0E [3], [5] :0;81@>2:0 >@5=F0 ?@8=F8?80;L=> =5 <>65B
40BL >1JOA=5=85 :2078AB0B8G5A:8< O2;5=8O< M;5:B@>48=0<8:8.  =59 M=5@38O ?>;O
A:0;O@=>3> ?>B5=F80;0 >:07K205BAO >B@8F0B5;L=>9, GB> ?@>B82>@5G8B 4065 70:>=C
C;>=0, =5 3>2>@O > 70:>=0E <03=5B87<0.  @01>B0E A45;0= 2K2>4, GB>
:2078AB0B8G5A:85 O2;5=8O 4>;6=K >?8AK20BLAO A0<>AB>OB5;L=>9 3@C??>9
C@02=5=89.
2. !2O7L <564C A:0;O@=K< 8 25:B>@=K< ?>B5=F80;0<8 4286CI53>AO 70@O40
>?@545;O5BAO A>>B=>H5=85<, :>B>@>5 =5 4>;6=> =0@CH0BLAO A = Ćv / c2 . BAN40
A;54C5B, GB> 2 @0<:0E :C;>=>2A:>9 :0;81@>2:8, 345 divA = 0, B0:65 =52>7<>6=> =5
B>;L:> :>@@5:B=>5 >1JOA=5=85 <03=8B=KE O2;5=89, => 8 @5H5=85 ?@>1;5<K
M;5:B@><03=8B=>9 <0AAK.
"5>@5B8G5A:89 0=0;87 2>7<>6=>AB8 >1JOA=5=8O :2078AB0B8G5A:8E O2;5=89
M;5:B@>48=0<8:8 (DC=:F88 03@0=60, C@02=5=8O 42865=8O, 70:>=K A>E@0=5=8O 8 B.4.) 40= 2
@01>B5 [2]. @82545< =5:>B>@K5 @57C;LB0BK, =0 :>B>@K5 <K 1C45< >?8@0BLAO ?@8
>1JOA=5=88 M:A?5@8<5=B>2. >A:>;L:C ?@0:B8G5A:8 2A5 M:A?5@8<5=BK, ?@82545==K5 2 [1],
A2O70=K A 2708<>459AB285< B>:>2, <K ?@82545< 425 D>@<C;K, >?8AK20NI85
2708<>459AB285 42CE M;5<5=B0@=KE B>:>2.
8A. 1
@8 2708<>459AB288 2-E M;5<5=B0@=KE B>:>2 <K 8<55<:
1. !8;>2>5 2708<>459AB285 B>:>2
(I1dl1I2dl2 )
2 2
d F12 = -ź R12 = -d F21 (1.1)
3
4ĄR12
2. 708<>459AB285 B>:>2 G5@57 2@0I0NI85 <><5=BK
ź
2 2
d M12 = [I1dl1 � I2dl2] = -d M21 (1.2)
4ĄR12
3
0: 284=> 87 D>@<C; B@5B89 ?@8=F8? LNB>=0 (@025=AB2> 459AB28O ?@>B82>459AB28N)
2A5340 2K?>;=O5BAO. -B> A;54AB285 DC=:F88 03@0=60, :>B>@0O >:07K205BAO 8=20@80=B=>9
>B=>A8B5;L=> ?@5>1@07>20=8O 0;8;5O.
> A2>5<C ?>AB@>5=8N C@02=5=8O 42865=8O =5 7028AOB >B 2K1>@0 8=5@F80;L=>9 A8AB5<K
>BAG5B0. >;55 B>3>, 5A;8 4065 MD8@ ACI5AB2C5B, 53> A:>@>ABL B0:65 �2K?0405B 87
C@02=5=89 42865=8O.
2. -:A?5@8<5=BK . !830;>20
@82545< >?8A0=85 ?5@2>3> M:A?5@8<5=B0 [1]. @C385 M:A?5@8<5=BK (!830;>20 8 4@. ! 5, 6,
7, 8, 14, 15, 17, 19 8 8:>;0520 ! 2, 4, 20, 21, 23) O2;ONBAO 20@80F8O<8 =0 BC 65 B5<C.
-:A?5@8<5=B ! 1 [1]. &8B8@C5<:
�@8 ?>4:;NG5=88 B>:0 : ->1@07=><C ?@>2>4=8:C ?>A;54=89 ?@8E>48B 2 ?>ABC?0B5;L=>5
42865=85.  @0<:0E 8725AB=KE ?@54AB02;5=89 ?>4>1=>5 42865=85 2>7<>6=> B>;L:> ?@8
2708<>459AB288 ->1@07=>3> ?@>2>4=8:0 A A>1AB25==K< <03=8B=K< ?>;5<.
8A. 2
1JOA=5=85 >A=>2K205BAO =0 ?@54?>;>65=88, GB> <03=8B=>5 ?>;5  1>:>2KE CG0AB:>2
B>:0 1, 2 >:07K205B 402;5=85 =0 65AB:> A2O70==K9 A =8<8 CG0AB>: B>:0 3 ?@>2>4=8:0,
?>4 459AB285< :>B>@>3> ?>A;54=89 ?@8E>48B 2 ?>ABC?0B5;L=>5 42865=85, C2;5:0O 70
A>1>9 8 CG0AB:8 B>:0 1, 2 ->1@07=>3> ?@>2>4=8:0. @8 4;8=5 :>=BC@0 2 2-3 @070 1>;LH5
H8@8=K, =0 3 ?>@O4:0 <5=LH59 A8;>9 459AB28O <03=8B=>3> ?>;O  =5?>4286=>3>
?@>2>4=8:0 4 =0 CG0AB>: B>:0 3 ?>4286=>3> ->1@07=>3> ?@>2>4=8:0 <>6=> ?@5=51@5GL.
;O @07@5H5=8O ?@>B82>@5G89 A 70:>=0<8 <5E0=8:8 <?5@>< 1K;> 4>?CI5=>
ACI5AB2>20=85 ?@>4>;L=>9 A8;K FQ%, 459AB2CNI59 24>;L ?@>2>4=8:>2 1, 2, >4=0:>
ACI5AB2>20=85 40==>9 A8;K ?@>B82>@5G8B >A=>20< :;0AA8G5A:>9 M;5:B@>48=0<8:8.
>6=> >1JOA=8BL MB> O2;5=85 G5@57 2708<>459AB285 ?@>2>4=8:>2 (1.1) 8;8 65 A
M=5@35B8G5A:>9 B>G:8 7@5=8O. K @0AA<>B@8< >10 ?>4E>40, :>B>@K5 ?@82>4OB :
>48=0:>2K< :0G5AB25==K< 8 :>;8G5AB25==K< @57C;LB0B0<.
1. -=5@35B8G5A:89 ?>4E>4. 0AA<>B@8< ?;>A:89 70<:=CBK9 :>;LF52>9 :>=BC@
@048CA>< R , >1@07>20==K9 ?@>2>4>< @048CA>< r. CABL 24>;L MB>3> :>=BC@0 B5G5B
B>: I.
>4AG8B05< M=5@38N ?>;O 25:B>@=>3> ?>B5=F80;0, A>740205<CN B>:><. =0 @02=0
2
W = LI / 2
=4C:B82=>ABL MB>3> :>=BC@0 A B>G=>ABLN 4> G;5=>2 ?>@O4:0 (r / R)4 @02=0 [6]
8R 7
L = ź0[R(ln - )
r 4
4
8A. 3
0 G0AB8 :>=BC@0 1C4CB 459AB2>20BL @0ABO3820NI85 A8; A ?>3>==>9 25;8G8=>9 f . =8
1C4CB A>74020BL =0BO65=85 ", AB@5<OI55AO �@0ABO=CBL :>=BC@, @07>@20BL 53>.
>?CAB8<, GB> 2 @57C;LB0B5 MB>3> :>=BC@ C25;8G8; A2>9 @048CA =0 25;8G8=C "R. ">: ?@8
MB>< =5 87<5=8BAO. 7<5=8BAO 8=4C:B82=>ABL :>=BC@0 8, A>>B25BAB25==> 53> M=5@38O.
7<5=5=85 M=5@388 5ABL @01>B0, A>25@H5==0O A8;0<8 f.
B0:
2 2
"A = [L(R + "R) - L(R)]I / 2 = "LI / 2 = 2ĄRf �" "R = 2Ą"RT
BAN40 <>6=> =09B8 25;8G8=C ?>3>==>9 A8;K
2 2 2
"A I "L I dL I 8R 3
f = = H" = ź0[ln - ]
"R 4ĄR"R 4ĄRdR 4ĄR r 4
!>>B25BAB25==>
2
I 8R 3
T = ź0[ln - ]
2 r 4
"0:8< >1@07><, =8G53> �AB@0==>3> 2 MB8E M:A?5@8<5=B0E =5 >1=0@C68205BAO. A5 ?;>A:85
70<:=CBK5 :>=BC@0 4>;6=K @0ABO3820BLAO ?@8 ?@>E>645=88 ?> =8< B>:0.
 MB>< A<KA;5 M:A?5@8<5=BK ! 5 8 6 �=5 2?8AK20NBAO 2 MB> >1JOA=5=85 (?@>B82>@5G0B
5<C). 81> 02B>@K 40;8 =5 ?>;=>3> >?8A0=8O M:A?5@8<5=B0, ;81> >= ?@>2545= =5:>@@5:B=>,
;81> 40=> =525@=>5 >1JOA=5=85.
2. !8;>2>9 ?>4E>4. @>2>4=8:8 :>=BC@0 1-2-3 (@8A. 2) >1@07CNB 65AB:CN A8AB5<C,
:>B>@0O 2708<>459AB2C5B A ?@>2>4=8:>< 4.  A>>B25BAB288 A D>@<C;>9 (2.1) <K
<>65< @0AA<0B@820BL ?0@=K5 2708<>459AB28O G0AB59 :>=BC@0 1-2-3 A ?@>2>4=8:>< 4.
02=>459AB2CNI0O MB8E A8; AB@5<8BLAO �>B>428=CBL ?@>2>4=8: 4 >B >AB0;L=>9
G0AB8 MB>3> :>=BC@0. 45AL B@5B89 70:>= LNB>=0 =5 =0@CH05BAO. 'B> :0A05BAO
<><5=B>2 A8;, B> MB8 <><5=BK 2708<=> C=8GB>60NBAO.
8A. 4
5
">G=> B0: 65 <>6=> @0AA<>B@5BL 8 >1JOA=8BL 4@C385 M:A?5@8<5=BK MB>9 3@C??K.
3. -:A?5@8<5=B . 8:>;0520
-:A?5@8<5=BK, @0AA<>B@5==K5 2KH5, >B=>A8;8AL : B0:><C :;0AAC, :>340 42865=85
?@>2>4=8:0 ?@>8AE>48;> 2 =0?@02;5=88, ?5@?5=48:C;O@=>< A0<><C ?@>2>4=8:C. 45AL <K
@0AA<>B@8< M:A?5@8<5=BK, :>340 42865=85 ?@>2>4=8:0 :>;;8=50@=> 53> >@85=B0F88.
-:A?5@8<5=B !3 [1]. ?8A0=85. �;O 45<>=AB@0F88 2K?>;=8<>AB8 70:>=>2 <5E0=8:8 ?@8
2708<>459AB288 ?5@?5=48:C;O@=KE M;5<5=B>2 B>:0 ?>4286=K9 ?@O<>;8=59=K9 ?@>2>4=8:
1 =0 ?>425A5 @07<5I05BAO =0 @0AAB>O=88 2 4 << >B >AB0;L=KE ?@>2>4=8:>2
?@O<>C3>;L=>3> :>=BC@0. <:>ABL ! 70@O605BAO 4> 10-20 :. @8 ?@>1>5 ?@><56CB:>2
<564C ?>4286=K< ?@>2>4=8:>< 1 8 ?@>2>4=8:0<8 :>=BC@0 ?>4286=K9 ?@>2>4=8:
?@8E>48B 2 ?>ABC?0B5;L=>5 42865=85 24>;L =0?@02;5=8O B>:0 2 =5< 2 =0?@02;5=88
459AB2CNI59 =0 =53> ?@>4>;L=>9 A8;K F . >?5@5G=K5 A8;K F @50:F88 >B ?>4286=>3>
Q% 4%
?@>2>4=8:0 1 ?@8;>65=K : 1>:>2><C ?@>2>4=8:C 3 :>=BC@0.
8A. 5
 MB>< >1JOA=5=88 =5B >1JOA=5=8O ?@8G8= ?>O2;5=8O A8;. @8 1>;LH>< =0?@O65=88 =0
5<:>AB8 2>7=8:05B 1>;LH>9 8<?C;LA=K9 @07@O4=K9 B>:. A=>2=K< ?5@5=>AG8:>< 70@O40
O2;ONBAO M;5:B@>=K ?@>2>48<>AB8. @8 @07@O45 M;5:B@>=K 4286CBAO ?@>B82 =0?@02;5=8O
B>:0, A>74020O <>I=K9 <5E0=8G5A:89 8<?C;LA. !>>B25BAB25==>, 2 A8;C 70:>=0 A>E@0=5=8O
8<?C;LA0 ?>;>68B5;L=K5 8>=K ?@>2>4=8:0 4>;6=K 42830BLAO 2 >1@0B=>< =0?@02;5=88. >
MB>9 ?@8G8=5 8E 42865=85 (B.5. 42865=85 ?@>2>4=8:0) A>2?0405B A =0?@02;5=85< B>:0.
4. -:A?5@8<5=B '5@=8:>20
-:A?5@8<5=B ! 40 [1]. ?8A0=85. �0 ?@>2>4=8: AB>:>< 2 <03=8B=>< ?>;5 ?>AB>O==>3>
<03=8B0 459AB2C5B A8;0 >@5=F0. 4=0:> 5A;8 ?@>2>4=8: 70:@KBL F8;8=4@8G5A:8<
M:@0=>< 87 <03=8B><O3:>3> <0B5@80;0, B> 459AB285 =0 ?@>2>4=8: <03=8B=>3> ?>;O
?@0:B8G5A:8 8AG5705B, => 70B> A8;0 >:07K205BAO ?@8;>65==>9 B5?5@L : >15AB>G5==><C
M:@0=C. /2;5=85 >1JOA=8<> B>;L:> ?@8 CG5B5 2708<>459AB28O B>:>2 ?@>2>4=8:0 8
8=4CF8@>20==KE M:2820;5=B=KE B>:>2 M:@0=0 A ?>;O<8 25:B>@=>3> ?>B5=F80;0 2>
2=CB@5==59 ?>;>AB8 M:@0=0.
6
8A. 6
4=0:> 8<55BAO 8 4@C3>5 >1JOA=5=85. &8;8=4@8G5A:89 M:@0= �7018@05B =0 A51O A8;>2K5
2=5H=53> ;8=88 <03=8B=>3> ?>;O.  MB>< A<KA;5 ?@>2>4=8: A B>:>< >:07K205BAO ?>4
<5=LH8< 2>7459AB285< 2=5H=53> <03=8B=>3> ?>;O (MDD5:B M:@0=8@>20=8O).  A8;C MB>3>
�459AB285 =0 ?@>2>4=8: <03=8B=>3> ?>;O ?@0:B8G5A:8 8AG5705B. -B> >4=0 ?>;>28=0,
>1JOA=5=8O.
0AA<>B@8< 4@C3CN ?>;>28=C. >:@C3 ?@>2>4=8:0 A B>:>< ACI5AB2C5B =0?@O65==>ABL
<03=8B=>3> ?>;O . =0 C1K205B :0: R-1 ?> <5@5 C40;5=8O >B ?@>2>4=8:0.  <03=8B=><
M:@0=5 8=4C:F8O <03=8B=>3> ?>;O  1C45B ?@81;878B5;L=> 2 ź @07 1>;LH5, G5< 2 2>74CE5
21;878 M:@0=0.
=5H=55 <03=8B=>5 ?>;5 B>65 A>7405B A2>N 8=4C:F8N 2 MB>< M:@0=5.  @57C;LB0B5 2 >4=>9
?>;>28=5 M:@0=0 8=4C:F8O >:07K205BAO 1>;LH5, G5< 2 4@C3>9 (2 >4=>9 ?>;>28=5 2=5H=55 8
B>:>2>5 ?>;O A:;04K20NBAO, 0 2 4@C3>9 - 2KG8B0NBAO). -B> A>7405B 3@0485=B M=5@388 ?>;O,
:>B>@K9 2>7459AB2C5B =0 M:@0=, 70AB02;OO 53> ?5@5<5I0BLAO. @8G5< A8;0 2>7459AB28O
>:07K205BAO 2> <=>3> @07 1>;LH5, G5< A8;0 2>7459AB28O =0 ?@>2>4=8: 157 M:@0=0.
A;8 1K =5 1K;> MB>3> O2;5=8O, B.5. 5A;8 1K A8;0 459AB2>20;0 =5 =0 M:@0=, 0
=5?>A@54AB25==> =0 ?@>2>4=8: A B>:><, B> M;5:B@><>B>@K 8 35=5@0B>@K >:070;8AL 1K
=54>;3>25G=K<8. 7-70 ?5@5<5==>3> 402;5=8O =0 ?@>2>4=8: 53> 87>;OF8O =5 <>3;0 1K
4>;3> A;C68BL (8AB8@0=85 87>;OF88 C M;5:B@><>B>@>2, 35=5@0B>@>2, B@0=AD>@<0B>@>2, @5;5
8 B.4.).
5. -:A?5@8<5=B !>;C=8=0 8 >AB8=0
-:A?5@8<5=B ! 33 [1]. �;O 45<>=AB@0F88 O2;5=8O 2708<>459AB28O 4286CI53>AO 70@O40 A
?>;5< 25:B>@=>3> ?>B5=F80;0  =0 M;5:B@>==>-;CG52CN B@C1:C 1 2 <5AB5 @0A?>;>65=8O
>B:;>=ONI8E ?;0AB8= 2 >45B0 B>@>840;L=0O >1<>B:0 3. ">@>840;L=0O >1<>B:0
2K?>;=5=0 87 =0@C6=>3> 8 2=CB@5==53> A;>52, =0<>B0===KE <54=K< ?@>2>4>< 0.62 << A
>1I8< :>;8G5AB2>< 28B:>2 500. 5>1E>48<>ABL 42CEA;>9=>9 =0<>B:8 2K720=0 B5<,
GB>1K 8A:;NG8BL <03=8B=K5 ?>;O :>;LF52>3> B>:0 (>4=0 >1<>B:0 ;52>-28=B>20O, 4@C30O -
?@02>28=B>20O): 1<>B:8 2:;NG5=K B0:, GB>1K 8E <03=8B=K5 ?>B>:8 AC<<8@>20;8AL.
-;5:B@>=K 2 B@C1:5 CA:>@O;8AL @07=>ABLN ?>B5=F80;>2 400 . 0 25@B8:0;L=K5
?;0AB8=K ?>4020;>AL ?>AB>O==>5; >B:;>=ONI55 =0?@O65=85 4;O 7040=8O 1078A=>3>
A<5I5=8O M;5:B@>==>3> ;CG0 =0 M:@0=5 (5-20 <<). ">: 2 >1<>B:5 <5=O;AO 2 ?@545;0E 0-5 .
7
8A. 7
57C;LB0BK M:A?5@8<5=B0 ?@54AB02;5=K =0 3@0D8:5. @8 C25;8G5=88 B>:0 >4=>3>
=0?@02;5=8O C3>; >B:;>=5=8O M;5:B@>==>3> ;CG0 C25;8G8205B A2>N 25;8G8=C ?> >B=>H5=8N
: 1078A=><C >B:;>=5=8N. #25;8G5=85 C3;0 >B:;>=5=8O M;5:B@>==>3> ;CG0 ?@8 =587<5==><
=0?@O65=88 =0 >B:;>=ONI8E ?;0AB8=0E >1CA;>2;5=> C<5=LH5=85< A:>@>AB8 42865=8O
M;5:B@>=>2 ?CG:0 70 AG5B 2708<>459AB28O 8E A ?>;5< 25:B>@=>3> ?>B5=F80;0 
B>@>840;L=>9 >1<>B:8. @8 87<5=5=88 B>:0 2 >1<>B:5 =0 >1@0B=K9 C3>; >B:;>=5=8O
M;5:B@>==>3> ;CG0 C<5=LH05B A2>N 25;8G8=C ?> >B=>H5=8N : 53> 1078A=><C >B:;>=5=8N,
@538AB@8@CO MDD5:B C25;8G5=8O A:>@>AB8 M;5:B@>=>2 ?CG:0 ?@8 8E 2708<>459AB288 A
?>;5< 25:B>@=>3> ?>B5=F80;0  B>@>840;L=>9 >1<>B:8.
"0:8< >1@07><, ?>;>68B5;L=K<8 @57C;LB0B0<8 >?8AK205<>3> >?KB0 >4=>7=0G=>
4>:07K205BAO ACI5AB2>20=85 >1KG=>3> :;0AA8G5A:>3> 0=0;>30 8725AB=>3> >?KB0
@>=>20-><0 8 ACI5AB2>20=85 MDD5:B0 87<5=5=8O A:>@>AB8 42865=8O M;5:B@>=>2 ?@8
8E 2708<>459AB288 A ?>;5< 25:B>@=>3> ?>B5=F80;0 . >;>68B5;L=K<8 @57C;LB0B0<8
>?KB0 >4=>7=0G=> ?>4B25@6405BAO B0:65 ACI5AB2>20=85 =58725AB=>3> @0=55 2 =0C:5
O2;5=8O ?@>4>;L=>3> <03=8B=>3> 2708<>459AB28O.
-B> 25AL<0 8=B5@5A=K9 M:A?5@8<5=B.  A>60;5=8N, 8AB>G=8:8, =0 :>B>@K5 AAK;05BAO 02B>@
@01>BK [1] =09B8 =5 C40;>AL, 0 87 >?8A0=8O B@C4=> AC48BL > :>@@5:B=>AB8 M:A?5@8<5=B0 8
53> 8=B5@?@5B0F88. 865 <K ?@54;>68< 4@C3>9 20@80=B M:A?5@8<5=B0, @>4AB25==>3>
M:A?5@8<5=BC !>;C=8=0 8 >AB8=0, 8 ?@54;0305< 53> ?>AB028BL.
@54;0305<K9 M:A?5@8<5=B. !E5<0 M:A?5@8<5=B0 ?@54AB02;5=0 =0 @8A. 8.
0 ?>425A5 CAB0=>2;5= 48A:, A>548=5==K9 A:>;L7OI8<8 :>=B0:B0<8 A 8AB>G=8:>< ?8B0=8O.
> 48A:C ?@>B5:05B B>: I2, :>B>@K9 2708<>459AB2C5B A <03=8B=K< ?>;5< B>=:>9 :0BCH:8
(:>;LF0 A N 28B:0<8) 8 <>65B ?>2>@0G820BLAO 2>:@C3 A2>59 >A8. =>3>28B:>20O :0BCH:0
70?8BK205BAO B>:>< I1.  @57C;LB0B5 2708<>459AB28O 48A:0 A :0BCH:>9 =0 48A: 4>;65=
459AB2>20BL 2@0I0NI89 <><5=B, ?>2>@0G820NI89 48A: =0 >?@545;5==K9 C3>;..
ź
2 2
d M12 = [I1dl1 � I2dl2] = -d M21 (1.2)
4ĄR12
5;8G8=0 <><5=B0 A8;  (= C3;0 >B:;>=5=8O) 7028A8B >B @0AAB>O=8O 0, :0: ?>:070=> =0 @8A.
9
8
8A. 8
 ?@8=F8?5 B>=:CN :0BCH:C <>6=> 70<5=8BL <03=8B><. !E5<0 MB> ?@82545=0 =0 @8A. 10.
8A. 9 8A. 10
9
6. #=8?>;O@=0O 8=4C:F8O 8 <>B>@ $0@045O
 MBC 3@C??C 2AE>4OB M:A?5@8<5=BK ! 37 (?KB . >48=0), 38 (C=8?>;O@=K9 <>B>@
$0@045O), 39 (C=8?>;O@=K9 35=5@0B>@) [1].
"5>@8O C=8?>;O@=>3> 35=5@0B>@0 45B0;L=> @07>1@0=0 8 87;>65=0 2 @01>B5 [2]. 45AL =5B
=5>1E>48<>AB8 ?5@5A:07K20BL 55 A>45@60=85. > 5ABL =5A:>;L:> 206=KE <><5=B>2,
:>B>@K5 A;54>20;> 1K 745AL @0AA<>B@5BL.
5@2K9 2>?@>A: 2@0I05BAO ;8 <03=8B=>5 ?>;5 2<5AB5 A <03=8B>< 2 C=8?>;O@=><
35=5@0B>@5 8;8 65 >=> ?>:>8BAO, E>BO <03=8B 2@0I05BAO? @8G8=0 ?>O2;5=8O MB>3> 2>?@>A0
2 B><, GB> -! =5 7028A8B >B A:>@>AB8 2@0I5=8O <03=8B0, B.5. <03=8B >AB05BAO :0: 1K
�157CG0AB=K< : 2708<>459AB28N. .. 0=40C AG8B0;, GB> ?>;5 42865BAO 2<5AB5 A
<03=8B>< (.. 0=40C, .. 8DH8F, �"5>@8O ?>;O). .. "0<< 8<5; 4@C3CN B>G:C
7@5=8O. = =0AB0820;, GB> <03=8B=>5 ?>;5 =5?>4286=>, 4065 5A;8 <03=8B 2@0I05BAO (..
"0<< �A=>2K B5>@88 M;5:B@8G5AB20). <5==> :=830 "0<<0 AD>@<8@>20;0 C <=>38E
>H81>G=CN B>G:C 7@5=8O.
>?@>1C5< @5H8BL ?@>AB5=L:CN 7040G:C. CABL 8<55BAO 15A:>=5G=K9 AB5@6=52>9 <03=8B
?@O<>C3>;L=>3> A5G5=8O (@8A. 11), >@85=B8@>20==K9 24>;L >A8 E.
8A. 11
>?CAB8<, MB>B <03=8B 70:@KB >B 0A ?;0AB8:><. >6=> ;8 >?@545;8BL: 42865BAO ;8
<03=8B 24>;L >A8 E 8;8 65 >= =5?>42865=?
'B>1K 40BL >B25B =0 MB>B 2>?@>A M:A?5@8<5=B0B>@ <>65B ?>AB028BL B0:>9 M:A?5@8<5=B. =
<>65B =0?>;=8BL ?;0AB<0AA>2K9 B078: B@0=AD>@<0B>@=K< <0A;>< 8 27OBL ?5=>?;0AB>2K5
:@C6:8 A 70:@5?;5==K<8 =0 =8E A25@EC <5B0;;8G5A:8<8 H0@8:0<8. 0B5< >= <>65B
70@O48BL MB8 H0@8:8 @07=K<8 70@O40<8 8 ?><5AB8BL 8E 2 B078:, @0A?>;>682 B078: =04
B5;>< <03=8B0. A;8 <03=8B=>5 ?>;5 =5?>4286=>, :@C6:8 1C4CB ?;020BL, @0A?>;030OAL
A;CG09=K< >1@07><. > 5A;8 <03=8B=>5 ?>;5 42865BAO, B> ?@>87>945B @0745;5=85 70@O4>2.
@C6:8 A ?>;>68B5;L=K<8 70@O40<8 A>15@CBAO 2 >4=>9 AB>@>=5, 0 A >B@8F0B5;L=K<8 - =0
?@>B82>?>;>6=>9 >B =8E AB>@>=5!
A;8 ?@8 42865=88 <03=8B0 53> <03=8B=>5 ?>;5 =5?>4286=>, :0:0O A8;0 8E A<>3;0
@0745;8BL? 03=8B=>5 ?>;5 5ABL A2>9AB2> <03=8B0. K =5 A<>65< <03=8B=>5 ?>;5 8
�65;57:C <03=8B0 @07=5AB8 2 @07=K5 AB>@>=K, A45;0BL 8E =57028A8<K<8 4@C3 >B 4@C30.
B>@>9 2>?@>A <K AD>@<C;8@C5<, ?>A;5 @0AA<>B@5=8O M:A?5@8<5=B0 >48=0.
-:A?5@8<5=B ! 37 [1]. ?8A0=85. �1=0@C65=>, GB> @50:F8O =0 F8;8=4@8G5A:><
<03=8B5-AB0B>@5 ?@8 2@0I0NI5<AO 48A:5-@>B>@5 2 C=8?>;O@=>< 42830B5;5 ?>;=>ABLN
>BACBAB2C5B.  @0<:0E 8725AB=KE ?@54AB02;5=89 O2;5=85 =5 8<55B :>@@5:B=>3>
>1JOA=5=8O, B0: :0: =0E>48BAO 2 ?@>B82>@5G88 A 70:>=0<8 <5E0=8:8.
 459AB28B5;L=>AB8 : <03=8BC ?@8;>65=K A:><?5=A8@>20==K5 ?@>4>;L=K5 A8;K FQ% >B
2@0I0NI53>AO 48A:0 8 =5?>4286=>3> ?@>2>4=8:0 B>:>?>42>40, 2 @57C;LB0B5 G53>
AC<<0@=K9 <><5=B =0 <03=8B5 @025= =C;N 8 >= >AB05BAO 2 A>AB>O=88 ?>:>O. >;L
10
AB0B>@0 2K?>;=O5B =5?>4286=K9 ?@>2>4=8: B>:>?>42>40, =0 :>B>@K9 ?5@5405BAO
@50:F8O >B <03=8B0 - ?>?5@5G=0O A8;0 F4%, >4=0:> =5?>A@54AB25==>3> 459AB28O =0
2@0I0NI89AO 48A:-@>B>@ <03=8B=>5 ?>;5 B>:>?>42>4OI53> ?@>2>4=8:0-AB0B>@0 =5
>:07K205B. "0:8< >1@07><, >B B>:>?>42>4OI53> ?@>2>4=8:0-AB0B>@0 2@0I0NI89AO
<><5=B ?5@5405BAO =0 <03=8B, 0 >B <03=8B0, 2 A2>N >G5@54L, 2@0I0NI89AO <><5=B
?5@5405BAO =0 48A:-@>B>@, ?@8 MB>< <03=8B 2K?>;=O5B @>;L 0:B82=>3> ?5@540B>G=>3>
B5;0, >AB020OAL 2A5 2@5<O =5?>4286=K<. !C<<0@=K9 2@0I0NI89 <><5=B =0 <03=8B5
2A5340 >AB05BAO @02=K< =C;N.
8A. 12
B>@>9 2>?@>A: A5340 ;8 AC<<0@=K9 2@0I0NI89 <><5=B =0 <03=8B5 >AB05BAO @02=K<
=C;N?
>?@>A MB>B 4>AB0B>G=> A;>6=K9 8 206=K9, ?>A:>;L:C >= A2O70= A A>1;N45=85< 3
?@8=F8?0 LNB>=0 (@025=AB2> 459AB28O ?@>B82>459AB28N). K @0AA<>B@8< A;CG09
?@O<>;8=59=>3> 42865=8O, ?>A:>;L:C =8:0:8E ?@8=F8?80;L=KE >B;8G89 2 @01>B5
C=8?>;O@=KE <>B>@>2 8 35=5@0B>@>2 >B ;8=59=KE <>B>@>2 8 35=5@0B>@>2 =5B.
8A. 13
0: 284=> 87 @8AC=:0 2 :>=BC@5 abcd ?@>B5:05B B>:. 0 B>:, ?@>B5:0NI89 ?> 4286CI59AO
?;0AB8=5 >B b : a, A> AB>@>=K <03=8B0 459AB2C5B A8;0 F1, =0?@02;5==0O 24>;L A:>@>AB8
11
?;0AB8=K. ">G=> B0:0O 65 A8;0, => =0?@02;5==0O 2 ?@>B82>?>;>6=CN AB>@>=C, 459AB2C5B =0
4286CI89AO <03=8B. 0 =5?>4286=CN ?5@5<KG:C cd A> AB>@>=K <03=8B0 B>65 459AB2C5B
A8;0, ?>A:>;L:C >B d : c ?@>B5:05B B>B 65 B>:. ">G=> B0:0O 65 A8;0, => =0?@02;5==0O 2
?@>B82>?>;>6=CN AB>@>=C, 459AB2C5B =0 4286CI89AO <03=8B.  @57C;LB0B5 =0 <03=8B
2>7459AB2C5B @07=>AB=0O 25;8G8=0 " F, =0?@02;5==0O 24>;L A:>@>AB8 <03=8B0 (@8A. 13).
=0 @02=0: " F = F1 - F2 . >A:>;L:C F1 = F2, A8;>2>5 2>7459AB285 =0 <03=8B 1C45B
459AB28B5;L=> >BACBAB2>20BL.
0:;NG5=85
K @0AA<>B@5;8 =5 2A5 M:A?5@8<5=BK, 87;>65==K5 2 @01>B5 [1]. ;O 0=0;870 >AB02H8EAO
M:A?5@8<5=B>2 C =0A =5B =5>1E>48<>9 8=D>@<0F88. "5< =5 <5=55, <>6=> A45;0BL
>?B8<8AB8G5A:89 2K2>4. A;8 ?@8 >?8A0=88 :2078AB0B8G5A:8E O2;5=89 M;5:B@>48=0<8:8
>?8@0BLAO =0 <5E0=8:C LNB>=0 (0 =5 =0 @5;OB828ABA:85 �D0=B0788), <>6=> A CA?5E><
40BL :>@@5:B=>5 >1JOA=5=85 ACI5AB2CNI8< �?0@04>:A0;L=K< M:A?5@8<5=B0;L=K<
@57C;LB0B0<.
AB>G=8: 8=D>@<0F88:
1. .. 8:>;052 !>2@5<5==0O M;5:B@>48=0<8:0 8 ?@8G8=K 55 ?0@04>:A0;L=>AB8.
http://www.trinitas.ru/rus/doc/0231/004a/02310011.htm
2. .. C;838=, .. C;838=0, .. >@=520. @878A @5;OB828ABA:8E B5>@89 ('0ABL 5,6)
http://n-t.ru/tp/ns/krt.htm
3. .. C;838=, .. C;838=0, .. >@=520. 52878O B5>@5B8G5A:8E >A=>2 @5;OB828ABA:>9
M;5:B@>48=0<8:8. http://n-t.ru/tp/ns/rt.htm
4. .. C;838=. -;5:B@>48=0<8:0 >B25@305B B5>@8N >B=>A8B5;L=>AB8.
new-idea.kulichki.net/?mode=physics
5. .. >@=520, .. C;838=, .. C;838=0. 0B5<0B8G5A:85 ;O?K 2 M;5:B@>48=0<8:5.
www.inauka.ru/http:/article78145/print.html
6. .. 0;0=B0@>2, .. &59B;8=. !'" #"!". (!?@02>G=0O :=830), - /. .
1970.