Лыков С.Н., Гасумянц В.Э., Рыков С.А. Квантовая механика. Учебное пособие
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C ɜɟɥɢɤɚ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɤɜɚɧɬɨɜɨ-ɦɟɯɚɧɢɱɟɫɤɨɣ ɚɦɩɥɢɬɭɞɨɣ «ɧɭɥɟɜɵɯ
ɤɨɥɟɛɚɧɢɣ»
0
/ Zm= , ɢ ɜɬɨɪɵɦ ɫɥɚɝɚɟɦɵɦ ɜ (20) ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ. ɉɪɢ ɷɬɨɦ
1!!n , ɬɚɤ ɱɬɨ ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɜɟɥɢɱɢɧɵ ɤɜɚɧɬɨɜɵɯ ɮɥɭɤɬɭɚɰɢɣ ɱɢɫɥɚ ɢ,
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɷɧɟɪɝɢɢ ɦɚɥɵ. ȼɫɟ ɷɬɨ, ɜ ɫɨɜɨɤɭɩɧɨɫɬɢ ɫ ɪɟɡɭɥɶɬɚɬɨɦ (10),
ɩɨɡɜɨɥɹɟɬ ɫɤɚɡɚɬɶ, ɱɬɨ ɜ ɤɨɝɟɪɟɧɬɧɨɦ ɫɨɫɬɨɹɧɢɢ
n
H
D ɫ 1!!D ɩɨɜɟɞɟɧɢɟ
ɤɜɚɧɬɨɜɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ ɜ ɧɚɢɛɨɥɶɲɟɣ ɫɬɟɩɟɧɢ ɧɚɩɨɦɢɧɚɟɬ ɤɚɪɬɢɧɭ
ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɤɨɥɟɛɚɧɢɹ ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ.
6.10. ɚ) ɋɤɚɥɹɪɧɨ ɭɦɧɨɠɚɹ ɥɟɜɭɸ ɢ ɩɪɚɜɭɸ ɱɚɫɬɢ ɩɪɢɜɟɞɟɧɧɵɯ ɜ ɭɫɥɨɜɢɢ
ɡɚɞɚɱɢ ɪɚɜɟɧɫɬɜ
c
ˆ
1 0 , c
ˆ
00 , (1)
ɧɚ
0,ɚ ɬɚɤɠɟ ɧɚ 1,ɧɚɯɨɞɢɦ ɜɢɞ ɨɩɟɪɚɬɨɪɚ c ɜ ɦɚɬɪɢɱɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ:
ˆ
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
00
10
ˆ
1110
0100
cc
cc
c , (2)
ɝɞɟ
ncnc
nn
c
c
ˆ
. Ɇɚɬɪɢɰɚ, ɷɪɦɢɬɨɜɨ ɫɨɩɪɹɠɟɧɧɚɹ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɦɚɬɪɢɰɟ
, ɢɦɟɟɬ ɜɢɞ:c
ˆ
¸
¸
¹
·
¨
¨
©
§
01
00
ˆ
c . (3)
Ⱦɟɣɫɬɜɭɹ ɷɬɨɣ ɦɚɬɪɢɰɟɣ ɧɚ ɫɬɨɥɛɰɵ, ɢɡɨɛɪɚɠɚɸɳɢɟ ɤɨɦɩɨɧɟɧɬɵ ɛɚɡɢɫɧɵɯ
ɜɟɤɬɨɪɨɜ,
¸
¸
¹
·
¨
¨
©
§
0
1
0 ,
¸
¸
¹
·
¨
¨
©
§
1
0
1 , (4)
ɩɨɥɭɱɚɟɦ:
c
ˆ
0 1 ,
c
ˆ
10 . (5)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɟɫɥɢ – ɨɩɟɪɚɬɨɪ, ɭɧɢɱɬɨɠɚɸɳɢɣ ɜɨɡɛɭɠɞɟɧɧɨɟ ɫɨɫɬɨɹɧɢɟ, ɬɨ
– «ɩɨɜɵɲɚɸɳɢɣ» ɨɩɟɪɚɬɨɪ: ɞɟɣɫɬɜɭɹ ɧɚ ɨɫɧɨɜɧɨɟ ɫɨɫɬɨɹɧɢɟ, ɨɧ ɩɨɪɨɠɞɚɟɬ
ɜɨɡɛɭɠɞɟɧɧɨɟ ɫɨɫɬɨɹɧɢɟ ɫɢɫɬɟɦɵ.
c
ˆ
c
ˆ
ɛ) ɉɟɪɟɦɧɨɠɟɧɢɟ ɦɚɬɪɢɰ (2) ɢ (3) ɞɚɟɬ:
c
ˆ
c
ˆ
¸
¸
¹
·
¨
¨
©
§
00
01
,
c
ˆ
c
ˆ
¸
¸
¹
·
¨
¨
©
§
10
00
. (6)
ɋɥɟɞɨɜɚɬɟɥɶɧɨ:
50
1
ˆˆˆˆ
}
ˆ
,
ˆ
{ {
cccccc . (7)
Ⱥɧɚɥɨɝɢɱɧɨ ɩɪɨɜɟɪɹɟɬɫɹ, ɱɬɨ c
ˆ
0
ˆ
c ,0
ˆˆ
cc , ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
0}
ˆ
,
ˆ
{ cc ,0}
ˆ
,
ˆ
{
cc . (8)
ɜ) ɂɡ (6) ɜɢɞɧɨ, ɱɬɨ ɨɩɟɪɚɬɨɪ
H c
ˆ
1
c
ˆ
ɜ ɛɚɡɢɫɟ 0, 1 ɢɦɟɟɬ
ɞɢɚɝɨɧɚɥɶɧɭɸ ɮɨɪɦɭ, ɩɪɢɱɟɦ ɫɨɫɬɨɹɧɢɹ
0 ɢ 1 ɩɪɢɧɚɞɥɟɠɚɬ ɫɨɛɫɬɜɟɧɧɵɦ
ɡɧɚɱɟɧɢɹɦ ɢ , ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɉɨɷɬɨɦɭ ɞɚɧɧɵɣ ɨɩɟɪɚɬɨɪ ɦɨɠɧɨ
ɬɪɚɤɬɨɜɚɬɶ ɤɚɤ ɝɚɦɢɥɶɬɨɧɢɚɧ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɵ:
0
0
H
1
H
1
ˆ
H H
c
ˆ
c
ˆ
. (9)
Ɉɩɟɪɚɬɨɪɵ ɢ , ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɟɫɹ ɫɜɨɣɫɬɜɚɦɢ (7)–(8), ɧɚɡɵɜɚɸɬ
ɮɟɪɦɢɨɧɧɵɦɢ ɨɩɟɪɚɬɨɪɚɦɢ ɭɧɢɱɬɨɠɟɧɢɹ ɢ ɪɨɠɞɟɧɢɹ; ɬɚɤɢɟ ɨɩɟɪɚɬɨɪɵ
ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɸɬɫɹ ɜ ɡɚɞɚɱɚɯ ɤɜɚɧɬɨɜɨɣ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ ɧɚɪɹɞɭ ɫ
ɛɨɡɨɧɧɵɦɢ ɨɩɟɪɚɬɨɪɚɦɢ ɢ , ɪɚɫɫɦɚɬɪɢɜɚɜɲɢɦɢɫɹ ɜ ɡɚɞɚɱɟ 6.7. Ɉ ɫɢɫɬɟɦɟ ɫ
ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ (9) ɢɧɨɝɞɚ ɝɨɜɨɪɹɬ ɤɚɤ ɨ «ɮɟɪɦɢɨɧɧɨɦ ɨɫɰɢɥɥɹɬɨɪɟ».
c
ˆ
c
ˆ
a
ˆ
a
ˆ
51
Ƚɥɚɜɚ 7. ɆɈɆȿɇɌ ɂɆɉɍɅɖɋȺ
ɉɨɧɹɬɢɟ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɜ ɤɜɚɧɬɨɜɨɣ ɬɟɨɪɢɢ ɩɨɥɧɨɫɬɶɸ ɫɜɹɡɚɧɨ ɫ
ɪɟɡɭɥɶɬɚɬɚɦɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɢɫɫɥɟɞɨɜɚɧɢɹ ɬɚɤ ɧɚɡɵɜɚɟɦɨɣ ɝɪɭɩɩɵ ɜɪɚɳɟɧɢɣ
(ɫɨɜɨɤɭɩɧɨɫɬɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɫɢɦɦɟɬɪɢɢ ɫɮɟɪɵ). ɉɪɢɦɟɧɟɧɢɸ ɜ ɤɜɚɧɬɨɜɨɣ
ɦɟɯɚɧɢɤɟ ɦɟɬɨɞɨɜ ɬɟɨɪɢɢ ɝɪɭɩɩ ɩɨɫɜɹɳɟɧɚ ɨɛɲɢɪɧɚɹ ɭɱɟɛɧɚɹ ɥɢɬɟɪɚɬɭɪɚ; ɧɢɠɟ
ɩɪɢɜɨɞɢɬɫɹ ɥɢɲɶ ɧɟɛɨɥɶɲɨɣ ɩɟɪɟɱɟɧɶ ɨɫɧɨɜɧɵɯ ɮɨɪɦɭɥ.
ɗɪɦɢɬɨɜɵɦɢ ɨɩɟɪɚɬɨɪɚɦɢ ɩɪɨɟɤɰɢɣ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ
, ɢ
ɨɩɪɟɞɟɥɹɸɬɫɹ ɭɧɢɬɚɪɧɵɟ ɨɩɟɪɚɬɨɪɵ ɩɨɜɨɪɨɬɨɜ ɧɚ ɩɪɨɢɡɜɨɥɶɧɵɟ ɭɝɥɵ
, ɢ ɜɨɤɪɭɝ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ:
x
J
ˆ
=
y
J
ˆ
=
z
J
ˆ
=
)(
ˆ
ĮR
x
D
y
D
z
D
R
ˆ
() ,
xx
eD
xx
Ji
e
ˆ
D
R
ˆ
(
yy
eD )
yy
Ji
e
ˆ
D
,
R
ˆ
(
zz
eD ) . (I)
zz
Ji
e
ˆ
D
ȼɟɤɬɨɪɧɵɣ ɨɩɟɪɚɬɨɪ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɢɦɟɟɬ ɜɢɞ
J
ˆ
x
e
x
J
ˆ
y
e
y
J
ˆ
z
e
z
J
ˆ
. (II)
ɋɤɚɥɹɪɧɵɣ ɨɩɟɪɚɬɨɪ ɤɜɚɞɪɚɬɚ ɦɨɦɟɧɬɚ ɨɩɪɟɞɟɥɟɧ ɮɨɪɦɭɥɨɣ
2222
ˆˆˆˆ
zyx
JJJ J . (III)
ȼɚɠɧɭɸ ɪɨɥɶ ɢɝɪɚɸɬ ɨɩɟɪɚɬɨɪɵ ɢ , ɷɪɦɢɬɨɜɨ ɫɨɩɪɹɠɟɧɧɵɟ ɩɨ
ɨɬɧɨɲɟɧɢɸ ɞɪɭɝ ɤ ɞɪɭɝɭ,
J
ˆ
J
ˆ
J
ˆ
x
J
ˆ
y
Ji
ˆ
, . (IV)
J
ˆ
x
J
ˆ
y
Ji
ˆ
ɋɩɪɚɜɟɞɥɢɜɵ ɫɥɟɞɭɸɳɢɟ ɤɨɦɦɭɬɚɰɢɨɧɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ:
zyx
JiJJ
ˆ
]
ˆ
,
ˆ
[ , , ; (V)
xzy
JiJJ
ˆ
]
ˆ
,
ˆ
[
yxz
JiJJ
ˆ
]
ˆ
,
ˆ
[
0]
ˆ
,
ˆ
[
2
x
JJ , , . (VI)0]
ˆ
,
ˆ
[
2
y
JJ 0]
ˆ
,
ˆ
[
2
z
JJ
ɂɡ (V) ɫɥɟɞɭɟɬ, ɱɬɨ ɧɟ ɫɭɳɟɫɬɜɭɟɬ ɤɜɚɧɬɨɜɵɯ ɫɨɫɬɨɹɧɢɣ, ɜ ɤɨɬɨɪɵɯ ɩɪɢ ɧɟ
ɪɚɜɧɨɦ ɧɭɥɸ ɦɨɦɟɧɬɟ ɜɫɟ ɬɪɢ ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ ɢɦɟɥɢ ɛɵ ɨɩɪɟɞɟɥɟɧɧɨɟ
ɡɧɚɱɟɧɢɟ; ɜɟɥɢɱɢɧɵ , ɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɧɟ ɢɡɦɟɪɢɦɵ. ɉɨɷɬɨɦɭ
ɫɨɫɬɨɹɧɢɹɦɢ
ɫ ɨɩɪɟɞɟɥɟɧɧɵɦ ɦɨɦɟɧɬɨɦ ɧɚɡɵɜɚɸɬ ɜɟɤɬɨɪɵ ɫɨɫɬɨɹɧɢɣ
x
J
y
J
z
J
mj,,
ɹɜɥɹɸɳɢɟɫɹ ɫɨɛɫɬɜɟɧɧɵɦɢ ɜɟɤɬɨɪɚɦɢ ɧɟ ɞɥɹ ɨɩɟɪɚɬɨɪɚ ɦɨɦɟɧɬɚ , ɚɞɥɹ
ɤɨɦɦɭɬɢɪɭɸɳɢɯ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɨɩɟɪɚɬɨɪɨɜ ɤɜɚɞɪɚɬɚ ɦɨɦɟɧɬɚ
ɢ ɩɪɨɟɤɰɢɢ
ɦɨɦɟɧɬɚ :
J
ˆ
2
ˆ
J
z
J
ˆ
2
ˆ
J
)1(, jjmj mj, , (VII)
52
z
J
ˆ
mmj , mj, , (VIII)
ɝɞɟ ) – ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɤɜɚɞɪɚɬɚ ɦɨɦɟɧɬɚ. ȿɫɥɢ ɜ ɤɚɱɟɫɬɜɟ
ɦɨɦɟɧɬɚ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɬɨɥɶɤɨ ɨɪɛɢɬɚɥɶɧɵɣ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɢɥɢ ɩɨɥɧɵɣ
ɦɨɦɟɧɬ (ɫ ɭɱɟɬɨɦ ɫɩɢɧɚ) ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɱɚɫɬɢɰɵ ɢɦɟɸɬ ɰɟɥɨɱɢɫɥɟɧɧɵɣ ɫɩɢɧ,
ɬɨ ɞɨɩɭɫɬɢɦɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɹɜɥɹɸɬɫɹ ɥɢɲɶ ɰɟɥɵɟ ɱɢɫɥɚ:
1(
jj
j
,...2,1,0 j . (IX)
ȿɫɥɢ ɠɟ ɱɚɫɬɢɰɵ ɢɦɟɸɬ ɩɨɥɭɰɟɥɵɣ ɫɩɢɧ, ɬɨ ɫɩɟɤɬɪ ɡɧɚɱɟɧɢɣ ɩɨɥɧɨɝɨ ɦɨɦɟɧɬɚ
ɫɨɞɟɪɠɢɬ ɤɚɤ ɰɟɥɵɟ ɬɚɤ ɢ ɩɨɥɭɰɟɥɵɟ ɱɢɫɥɚ:
j
,...2,
2
3
,1,
2
1
,0 j . (X)
ȼɟɥɢɱɢɧɚ ɧɚɡɵɜɚɟɬɫɹ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɨɣ ɦɨɦɟɧɬɚ ɜ ɫɨɫɬɨɹɧɢɹɯj
mj,.
ɉɪɢ ɡɚɞɚɧɧɨɦ ɡɧɚɱɟɧɢɢ ɩɪɨɟɤɰɢɹ ɦɨɦɟɧɬɚ (ɟɟ ɩɪɢɧɹɬɨ ɨɛɨɡɧɚɱɚɬɶ ɛɭɤɜɨɣ
) ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɣ ɫ ɲɚɝɨɦ, ɪɚɜɧɵɦ ɟɞɢɧɢɰɟ:
j
z
J
m
12 j
jjjm
,...,1, . (XI)
ȼ ɛɚɡɢɫɟ, ɨɛɪɚɡɨɜɚɧɧɨɦ ɫɨɫɬɨɹɧɢɹɦɢ
mj, ɫ ɡɚɞɚɧɧɵɦ , ɦɚɬɪɢɰɚ
ɨɩɟɪɚɬɨɪɚ ɞɢɚɝɨɧɚɥɶɧɚ; ɟɟ ɞɢɚɝɨɧɚɥɶɧɵɦɢ ɷɥɟɦɟɧɬɚɦɢ ɹɜɥɹɸɬɫɹ ɭɤɚɡɚɧɧɵɟ
ɜɵɲɟ ɡɧɚɱɟɧɢɹ . ɉɪɢ ɷɬɨɦ ɧɟ ɪɚɜɧɵɟ ɧɭɥɸ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɨɜ
ɢ ɦɨɠɧɨ ɧɚɣɬɢ ɩɨ ɮɨɪɦɭɥɚɦ
j
z
J
ˆ
m
x
J
ˆ
y
J
ˆ
1
ˆˆ
1 mJmmJm
xx
)1()()2/1( mjmj ,
1
ˆˆ
1 mJmmJm
yy
)1()()2/( mjmji , (XII)
ɚ ɦɚɬɪɢɰɵ ɨɩɟɪɚɬɨɪɨɜ ɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡ ɫɨɨɬɧɨɲɟɧɢɣ
J
ˆ
J
ˆ
J
ˆ
mj, )1()( mjmj 1, mj ,
J
ˆ
1, mj )1()( mjmj mj, . (XIII)
Ɉɪɛɢɬɚɥɶɧɵɣ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ ɨɩɢɫɵɜɚɟɬɫɹ ɜɟɤɬɨɪɧɵɦ
ɨɩɟɪɚɬɨɪɨɦ , ɞɟɣɫɬɜɭɸɳɢɦ ɧɚ ɜɨɥɧɨɜɭɸ ɮɭɧɤɰɢɸ. ȼ ɤɨɨɪɞɢɧɚɬɧɨɦ
ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɨɧ ɢɦɟɟɬ ɜɢɞ (ɜ ɟɞɢɧɢɰɚɯ ):
l
ˆ
=
u u
rprl i=/)
ˆ
(
ˆ
, (XIV)
53
ɢɥɢ, ɜ ɫɮɟɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ( M
T
,,
r
),
i l
ˆ
M
e i
Tw
w
T
e
Mw
w
Tsin
1
. (XV)
Ɉɩɟɪɚɬɨɪɵ ɩɪɨɟɤɰɢɣ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ
¸
¹
·
¨
©
§
w
w
w
w
y
z
z
yil
x
ˆ
,
¸
¹
·
¨
©
§
w
w
w
w
z
x
x
zil
y
ˆ
,
Mw
w
¸
¸
¹
·
¨
¨
©
§
w
w
w
w
i
x
y
y
xil
z
ˆ
(XVI)
ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɤɨɦɦɭɬɚɰɢɨɧɧɵɦ ɫɨɨɬɧɨɲɟɧɢɹɦ (V), (VI). Ɉɩɟɪɚɬɨɪ ɤɜɚɞɪɚɬɚ
ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ
2222
ˆˆˆ
ˆ
zyx
lll l
2
2
2
sin
1
sin
sin
1
Mw
w
T
Tw
w
T
Tw
w
T
(XVII)
ɦɨɠɟɬ ɛɵɬɶ ɜɵɪɚɠɟɧ ɱɟɪɟɡ ɭɝɥɨɜɭɸ ɱɚɫɬɶ ɨɩɟɪɚɬɨɪɚ Ʌɚɩɥɚɫɚ, ɬɚɤ ɤɚɤ ɜ
ɫɮɟɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɨɩɟɪɚɬɨɪ Ʌɚɩɥɚɫɚ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
w
w
w
w
'
r
r
r
r
2
2
1
Mw
w
T
Tw
w
T
Tw
w
T
2
2
222
sin
1
sin
sin
1
rr
2
2
2
11
r
r
r
r
r
w
w
w
w
2
ˆ
l . (XVIII)
ɋɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɨɩɟɪɚɬɨɪɚ ɹɜɥɹɸɬɫɹ ɱɢɫɥɚ
2
ˆ
l )1(
l
l
ɝɞɟ
l
–
ɚɛɫɨɥɸɬɧɚɹ ɜɟɥɢɱɢɧɚ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ. ɗɬɚ ɜɟɥɢɱɢɧɚ
ɩɪɢɧɢɦɚɟɬ ɬɨɥɶɤɨ ɰɟɥɨɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ:
...,2,1,0
l
.
ȼ ɫɥɭɱɚɟ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɫɮɟɪɢɱɟɫɤɢ ɫɢɦɦɟɬɪɢɱɧɨɦ ɩɨɥɟ,
)()(
r
U
U
r , ɝɚɦɢɥɶɬɨɧɢɚɧ
H
ˆ
ɢ ɨɩɟɪɚɬɨɪɵ , ɤɨɦɦɭɬɚɬɢɜɧɵ ɞɪɭɝ ɫ ɞɪɭɝɨɦ.
ɉɨɷɬɨɦɭ ɜɟɥɢɱɢɧɚ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɢ ɩɪɨɟɤɰɢɹ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɧɚ
ɨɞɧɨ ɩɪɨɢɡɜɨɥɶɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ, ɜɵɛɢɪɚɟɦɨɟ ɜ ɤɚɱɟɫɬɜɟ ɨɫɢ
, ɩɪɟɞɫɬɚɜɥɹɸɬ
ɫɨɛɨɣ ɫɨɯɪɚɧɹɸɳɢɟɫɹ ɜɟɥɢɱɢɧɵ (ɢɧɬɟɝɪɚɥɵ ɞɜɢɠɟɧɢɹ).
2
ˆ
l
z
l
ˆ
z
Ɋɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ ɒɪɟɞɢɧɝɟɪɚ ɫɨ ɫɮɟɪɢɱɟɫɤɢ ɫɢɦɦɟɬɪɢɱɧɵɦ
ɩɨɬɟɧɰɢɚɥɨɦ,
\
¸
¸
¹
·
¨
¨
©
§
' )(
2
2
rU
m
=
H \ )(r )(r\ ,
54
ɦɨɝɭɬ ɛɵɬɶ ɧɚɣɞɟɧɵ ɜ ɜɢɞɟ ɜɨɥɧɨɜɵɯ ɮɭɧɤɰɢɣ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ ɫ
ɨɩɪɟɞɟɥɟɧɧɨɣ ɜɟɥɢɱɢɧɨɣ
l
ɢ ɩɪɨɟɤɰɢɟɣ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ
l
l
l
m ,...,1,:
\ )(
r )(
r
R
),( MT
lm
Y , (XIX)
ɝɞɟ
r
r
rR
)(
)(
F
,
ɩɪɢɱɟɦ ɜɫɩɨɦɨɝɚɬɟɥɶɧɚɹ ɮɭɧɤɰɢɹ )(
r
F – ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ, ɩɨ ɮɨɪɦɟ
ɫɨɜɩɚɞɚɸɳɟɝɨ ɫ ɨɞɧɨɦɟɪɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɒɪɟɞɢɧɝɟɪɚ:
0)()(
2
)1(2)(
2
2
22
2
F
¸
¸
¹
·
¨
¨
©
§
H
F
rrU
rm
llm
rd
rd =
=
. (XX)
Ɏɭɧɤɰɢɢ )ɨɩɪɟɞɟɥɹɸɬɫɹ (ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɮɚɡɨɜɨɝɨ ɦɧɨɠɢɬɟɥɹ)
ɮɨɪɦɭɥɨɣ
,( MT
lm
Y
MT ),(
lm
Y
M
T
S
mi
m
l
m
eP
ml
mll
)(cos)1(
!)(
!)(
4
)12(
)(
, (XXI)
ɝɞɟ – ɩɪɢɫɨɟɞɢɧɟɧɧɵɟ ɩɨɥɢɧɨɦɵ Ʌɟɠɚɧɞɪɚ:)(cos
)(
T
m
l
P
l
ml
ml
m
l
m
l
d
d
l
P 1cos
)cos(
sin
!2
1
)(cos
2
)(
T
T
T T
. (XXII)
ɉɪɢɜɟɞɟɦ ɹɜɧɵɟ ɜɵɪɚɠɟɧɢɹ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɜɵɯ ɮɭɧɤɰɢɣ :),( MT
lm
Y
S
4
1
00
Y , T
S
cos
4
3
10
Y ,
1cos3
16
5
2
20
T
S
Y ,
Mr
r
T
S
i
eY sin
8
3
1,1
B ,
Mr
r
TT
S
i
eY sincos
8
15
1,2
B ,
Mr
r
T
S
i
eY
22
2,2
sin
32
15
.
ȼɨ ɦɧɨɝɢɯ ɡɚɞɚɱɚɯ ɩɨɬɟɧɰɢɚɥ )(
r
U
ɨɛɥɚɞɚɟɬ ɫɜɨɣɫɬɜɨɦ
0)(lim
2
rUr ɩɪɢ 0o
r
.
ȼ ɬɚɤɢɯ ɫɥɭɱɚɹɯ ɬɪɟɛɨɜɚɧɢɟ ɨɝɪɚɧɢɱɟɧɧɨɫɬɢ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ (XIX) ɩɪɢɜɨɞɢɬ
55
ɤ ɬɨɦɭ, ɱɬɨ «ɪɚɞɢɚɥɶɧɚɹ» ɮɭɧɤɰɢɹ )(
r
R
ɜɛɥɢɡɢ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɨɛɥɚɞɚɟɬ
ɫɥɟɞɭɸɳɢɦ ɩɨɜɟɞɟɧɢɟɦ:
l
l
rrR ~)(, 0o
r
. (XXIII)
ɉɪɢ ɷɬɨɦ )(
r
F ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɝɪɚɧɢɱɧɨɦɭ ɭɫɥɨɜɢɸ ɜɢɞɚ
0)0(
F . (XXIV)
ɉɪɢ
fo
r
ɩɨɜɟɞɟɧɢɟ )(
r
R
ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɟɦ, ɤ ɤɚɤɨɣ ɱɚɫɬɢ
ɷɧɟɪɝɟɬɢɱɟɫɤɨɝɨ ɫɩɟɤɬɪɚ ɨɬɧɨɫɹɬɫɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɫɨɫɬɨɹɧɢɹ. ȿɫɥɢ ɪɟɲɟɧɢɹ
ɭɪɚɜɧɟɧɢɹ (XX) ɩɪɢɧɚɞɥɟɠɚɬ ɞɢɫɤɪɟɬɧɵɦ ɭɪɨɜɧɹɦ ɷɧɟɪɝɢɢ , ɬɨ ɞɥɹ ɧɢɯ
ɫɩɪɚɜɟɞɥɢɜɚ ɬɚɤ ɧɚɡɵɜɚɟɦɚɹ
ɨɫɰɢɥɥɹɰɢɨɧɧɚɹ ɬɟɨɪɟɦɚ. ɋɨɝɥɚɫɧɨ ɷɬɨɣ ɬɟɨɪɟɦɟ
ɧɢɡɲɟɦɭ ɭɪɨɜɧɸ ɩɪɢɧɚɞɥɟɠɢɬ ɮɭɧɤɰɢɹ
FH
HF
, ɧɟ ɢɦɟɸɳɚɹ ɧɢ ɨɞɧɨɝɨ ɭɡɥɚ, ɚ
ɤɚɠɞɨɦɭ ɫɥɟɞɭɸɳɟɦɭ ɭɪɨɜɧɸ ɜ ɩɨɪɹɞɤɟ ɜɨɡɪɚɫɬɚɧɢɹ
H ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɪɟɲɟɧɢɟ
ɫ ɛɨɥɶɲɢɦ ɧɚ ɟɞɢɧɢɰɭ ɱɢɫɥɨɦ ɭɡɥɨɜ. Ɉɱɟɜɢɞɧɨ, ɮɭɧɤɰɢɢ
r
r
r
R
/
)()( F ɢ )(
r
F
ɢɦɟɸɬ ɨɞɧɨ ɢ ɬɨ ɠɟ ɱɢɫɥɨ ɭɡɥɨɜ; ɷɬɨ ɱɢɫɥɨ ɧɚɡɵɜɚɟɬɫɹ ɪɚɞɢɚɥɶɧɵɦ
ɤɜɚɧɬɨɜɵɦ ɱɢɫɥɨɦ. Ɏɭɧɤɰɢɹ )
r
n
()( rRrR
ln
r
{ , ɩɪɢɧɚɞɥɟɠɚɳɚɹ ɞɢɫɤɪɟɬɧɨɦɭ
ɭɪɨɜɧɸ ɷɧɟɪɝɢɢ
ln
r
H H , ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ
r
ɨɬ ɧɭɥɹ ɞɨ ɛɟɫɤɨɧɟɱɧɨɫɬɢ
ɫɨɜɟɪɲɚɟɬ ɨɫɰɢɥɥɹɰɢɣ ɢ ɡɚɬɟɦ ɦɨɧɨɬɨɧɧɨ ɭɛɵɜɚɟɬ ɬɚɤ, ɱɬɨ ɧɨɪɦɢɪɨɜɨɱɧɵɣ
ɢɧɬɟɝɪɚɥ ɞɥɹ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɨɤɚɡɵɜɚɟɬɫɹ ɫɯɨɞɹɳɢɦɫɹ ɢ ɦɨɠɟɬ ɛɵɬɶ ɜɵɛɪɚɧ
ɪɚɜɧɵɦ ɟɞɢɧɢɰɟ:
r
n
\
³
2
3
mln
r
d r :
³³
f
2
0
2
2
mlln
YdRrdr
r
1
0
2
2
³
f
ln
r
Rrdr .
Ɏɭɧɤɰɢɢ ɠɟ ), ɩɪɢɧɚɞɥɟɠɚɳɢɟ ɡɧɚɱɟɧɢɹɦ()(
rRrR
lk
{ mk 2
/
22
= H ɜ
ɧɟɩɪɟɪɵɜɧɨɣ ɱɚɫɬɢ ɷɧɟɪɝɟɬɢɱɟɫɤɨɝɨ ɫɩɟɤɬɪɚ, ɭɛɵɜɚɸɬ ɧɚ ɛɟɫɤɨɧɟɱɧɨɫɬɢ ɧɟ
ɦɨɧɨɬɨɧɧɨ ɢ ɧɟ ɫɬɨɥɶ ɛɵɫɬɪɨ, ɢɯ ɧɨɪɦɢɪɨɜɨɱɧɵɟ ɢɧɬɟɝɪɚɥɵ ɪɚɫɯɨɞɹɬɫɹ. ɗɬɢ
ɮɭɧɤɰɢɢ ɢɦɟɸɬ ɛɟɫɤɨɧɟɱɧɨ ɦɧɨɝɨ ɭɡɥɨɜ, ɩɪɢɱɟɦ ɨɫɰɢɥɥɹɰɢɢ ɩɪɢ)(rR
lk
fo
r
ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɞɥɢɧɨɣ ɜɨɥɧɵ O, ɤɨɬɨɪɚɹ, ɤɚɤ ɨɛɵɱɧɨ, ɫɜɹɡɚɧɚ ɫ
ɜɨɥɧɨɜɵɦ ɱɢɫɥɨɦ
k ɪɚɜɟɧɫɬɜɨɦ k
/
2S O .
Ɂɚɞɚɱɢ
7.1. Ⱦɥɹ ɫɨɫɬɨɹɧɢɹ mj, ɫ ɨɩɪɟɞɟɥɟɧɧɵɦ ɡɧɚɱɟɧɢɟɦ ɜɟɥɢɱɢɧɵ ɦɨɦɟɧɬɚ
ɢɦɩɭɥɶɫɚ ɢ ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ
jmJ
z
ɧɚɣɬɢ ɫɪɟɞɧɢɟ ɡɧɚɱɟɧɢɹ ɤɜɚɞɪɚɬɨɜ
ɩɪɨɟɤɰɢɣ ɦɨɦɟɧɬɚ ɧɚ ɞɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɧɵɟ ɨɫɢ.
56
7.2. Ⱦɥɹ ɫɨɫɬɨɹɧɢɹ mj, ɧɚɣɬɢ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ
ɧɚ ɩɪɨɢɡɜɨɥɶɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ, ɡɚɞɚɜɚɟɦɨɟ ɟɞɢɧɢɱɧɵɦ ɜɟɤɬɨɪɨɦ .
n
J
n
7.3. ɉɭɫɬɶ mj
~
,–ɫɨɫɬɨɹɧɢɟ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ ɜɟɥɢɱɢɧɨɣ ɦɨɦɟɧɬɚ ɢ
ɩɪɨɟɤɰɢɟɣ ɦɨɦɟɧɬɚ ɧɚ ɧɟɤɨɬɨɪɨɟ ɧɚɩɪɚɜɥɟɧɢɟ, ɡɚɞɚɧɧɨɟ ɟɞɢɧɢɱɧɵɦ
ɜɟɤɬɨɪɨɦ . ɇɚɣɬɢ ɭɫɪɟɞɧɟɧɧɵɣ ɜɟɤɬɨɪ ɦɨɦɟɧɬɚ
mJ
n
n J .
7.4. ɑɚɫɬɢɰɚ ɧɚɯɨɞɢɬɫɹ ɜ ɫɨɫɬɨɹɧɢɢ m,1 ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ ɩɪɨɟɤɰɢɟɣ
ɦɨɦɟɧɬɚ ; ɜɟɥɢɱɢɧɚ ɦɨɦɟɧɬɚ 1
mJ
z
j . ɇɚɣɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɪɚɡɥɢɱɧɵɯ
ɡɧɚɱɟɧɢɣ ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ ɧɚ ɧɨɜɭɸ ɨɫɶ
z
~
, ɫɨɫɬɚɜɥɹɸɳɭɸ ɫ ɩɟɪɜɨɧɚɱɚɥɶɧɨɣ
ɨɫɶɸ ɡɚɞɚɧɧɵɣ ɭɝɨɥ .
z T
7.5. Ⱦɥɹ ɫɜɨɛɨɞɧɨ ɞɜɢɠɭɳɟɣɫɹ ɱɚɫɬɢɰɵ ɧɚɣɬɢ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ
ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ ɜɟɥɢɱɢɧɨɣ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɢ
ɨɩɪɟɞɟɥɟɧɧɨɣ ɩɪɨɟɤɰɢɟɣ ɦɨɦɟɧɬɚ.
7.6. ɋɜɨɛɨɞɧɚɹ ɱɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɫ ɨɩɪɟɞɟɥɟɧɧɵɦ ɢɦɩɭɥɶɫɨɦ k
p
= ɜ
ɩɨɥɨɠɢɬɟɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ .
z
ɚ) Ɉɩɪɟɞɟɥɢɬɶ ɜɨɡɦɨɠɧɵɟ ɡɧɚɱɟɧɢɹ ɩɪɨɟɤɰɢɢ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɧɚ
ɨɫɶ .
z
ɛ) Ʉɚɤ ɡɚɜɢɫɢɬ ɨɬ k ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɱɚɫɬɢɰɚ ɛɭɞɟɬ ɨɛɥɚɞɚɬɶ
ɦɨɦɟɧɬɨɦ
l
(ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ)?
7.7. ɑɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɜ ɫɮɟɪɢɱɟɫɤɢ ɫɢɦɦɟɬɪɢɱɧɨɦ ɩɨɥɟ
(ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɣ ɨɫɰɢɥɥɹɬɨɪ). Ⱦɥɹ ɫɬɚɰɢɨɧɚɪɧɵɯ
ɫɨɫɬɨɹɧɢɣ ɜɢɞɚ ), ɩɪɢɧɚɞɥɟɠɚɳɢɯ ɭɪɨɜɧɸ ɷɧɟɪɝɢɢ, ɛɥɢɠɚɣɲɟɦɭ
ɤ ɨɫɧɨɜɧɨɦɭ ɷɧɟɪɝɟɬɢɱɟɫɤɨɦɭ ɭɪɨɜɧɸ, ɧɚɣɬɢ:
2/)(
22
0
rmU Z r
,,(
321
zyx
nnn
\
ɚ) ɍɫɪɟɞɧɟɧɧɵɣ ɜɟɤɬɨɪ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ l .
ɛ) ȼɟɪɨɹɬɧɨɫɬɶ ɡɧɚɱɟɧɢɣ ɜɟɥɢɱɢɧɵ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ
ml
W
l
ɢ
ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ .
ml
z
{
7.8. ɑɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɜ ɞɜɭɦɟɪɧɨɦ «ɤɭɥɨɧɨɜɫɤɨɦ» ɩɨɥɟ
r
U
D
)(r ,
22
yxr ,0!D .
ɇɚɣɬɢ ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ ɫɩɟɤɬɪ ɫɜɹɡɚɧɧɵɯ ɫɨɫɬɨɹɧɢɣ ɱɚɫɬɢɰɵ ɩɪɢ ɥɸɛɨɦ
ɞɨɩɭɫɬɢɦɨɦ ɡɧɚɱɟɧɢɢ ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ .
z
l
57
Ɋɟɲɟɧɢɹ
7.1. ɂɫɯɨɞɢɦ ɢɡ ɬɨɝɨ, ɱɬɨ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɨɩɟɪɚɬɨɪɨɜ
r
J
ˆ
r
x
J
ˆ
y
Ji
ˆ
(1)
ɜɟɤɬɨɪ ɫɨɫɬɨɹɧɢɹ
mj, ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ 1, rmjC ɢ ɩɨɷɬɨɦɭ, ɫ ɭɱɟɬɨɦ
ɜɡɚɢɦɧɨɣ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɫɨɫɬɨɹɧɢɣ
mj, ɫ ɪɚɡɥɢɱɚɸɳɢɦɢɫɹ ɡɧɚɱɟɧɢɹɦɢ ,
ɢɦɟɸɬ ɦɟɫɬɨ ɞɜɚ ɪɚɜɟɧɫɬɜɚ:
m
mj,( r
x
J
ˆ
y
Ji
ˆ
)
0, mj . (2)
ȼɵɞɟɥɢɜ ɜ (2) ɜɟɳɟɫɬɜɟɧɧɭɸ ɢ ɦɧɢɦɭɸ ɱɚɫɬɢ, ɢɦɟɟɦ:
x
J 0
y
J . (3)
ȼ ɫɢɥɭ ɭɩɨɦɹɧɭɬɨɝɨ ɨɛɫɬɨɹɬɟɥɶɫɬɜɚ ɫɩɪɚɜɟɞɥɢɜɵ ɬɚɤɠɟ ɪɚɜɟɧɫɬɜɚ
mj,( r
x
J
ˆ
y
Ji
ˆ
)
2
0, mj ,
ɬɨ ɟɫɬɶ
mj,( r
2
ˆ
x
J
2
ˆ
)
ˆˆˆˆ
(
yxyyx
JJJJJi ) 0, mj . (4)
Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ
mj,)
ˆˆˆˆ
(
xyyx
JJJJ 0, mj (5)
ɢ
mj,(
2
ˆ
x
J
2
ˆ
y
J ) 0, mj ,
ɬɨ ɟɫɬɶ
2
x
J
2
y
J . (6)
Ɂɚɦɟɬɢɦ ɬɟɩɟɪɶ, ɱɬɨ ɫ ɭɱɟɬɨɦ (6)
22
yx
JJ
2
2
x
J
2
2
y
J . (7)
ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ,
22
yx
JJ mj,(
2
ˆ
J
2
ˆ
z
J )
2
)1(, mjjmj . (8)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
2
x
J
22
)1(
2
1
mjjJ
y
. (9)
58
Ɉɬɦɟɬɢɦ, ɱɬɨ ɩɪɢ ɭɫɪɟɞɧɟɧɧɵɟ ɤɜɚɞɪɚɬɵ ɩɪɨɟɤɰɢɣ ɦɨɦɟɧɬɚ (9)
ɞɨɫɬɢɝɚɸɬ ɧɚɢɦɟɧɶɲɟɝɨ ɡɧɚɱɟɧɢɹ; ɨɧɨ ɪɚɜɧɨ 2
jm r
/
j ɢ ɧɟ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ ɩɪɢ
ɜɫɟɯ 0. ɉɨɫɤɨɥɶɤɭ ɷɬɨɬ ɮɚɤɬ ɢɦɟɟɬ ɦɟɫɬɨ ɧɚɪɹɞɭ ɫ ɪɚɜɟɧɫɬɜɚɦɢ (3), ɨɧ
ɭɤɚɡɵɜɚɟɬ ɧɚ ɩɪɢɫɭɬɫɬɜɢɟ «ɤɜɚɧɬɨɜɵɯ ɮɥɭɤɬɭɚɰɢɣ» ɜɟɥɢɱɢɧ ,.Ⱦɪɭɝɢɦɢ
ɫɥɨɜɚɦɢ, ɜ ɫɨɫɬɨɹɧɢɹɯ
!j
x
J
y
J
mj,,ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɯɫɹ ɨɩɪɟɞɟɥɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ
ɢ , ɜɟɥɢɱɢɧɵ ɢ ɧɟ ɢɦɟɸɬ ɨɩɪɟɞɟɥɟɧɧɵɯ ɡɧɚɱɟɧɢɣ.
2
J
z
J
x
J
y
J
Ⱦɟɣɫɬɜɭɹ 1ɪɚɡ ɨɩɟɪɚɬɨɪɨɦ ɧɚ ɨɛɟ ɫɬɨɪɨɧɵ ɪɚɜɟɧɫɬɜɚn
z
J
ˆ
z
J
ˆ
mmj , mj, (10)
ɢ ɫɤɚɥɹɪɧɨ ɭɦɧɨɠɚɹ ɪɟɡɭɥɶɬɚɬ ɧɚ
mj,,ɩɪɢɞɟɦ ɤ ɫɨɨɬɧɨɲɟɧɢɸ
n
z
nn
z
JmJ . (11)
ɉɨɞɨɛɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ ɫɪɟɞɧɢɦɢ ɡɧɚɱɟɧɢɹɦɢ (ɩɪɢ ɥɸɛɨɦ )
ɯɚɪɚɤɬɟɪɧɵ ɞɥɹ ɧɟ ɮɥɭɤɬɭɢɪɭɸɳɟɣ ɜɟɥɢɱɢɧɵ.
n
7.2. Ɉɩɟɪɚɬɨɪ ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ, ɡɚɞɚɜɚɟɦɨɟ
ɟɞɢɧɢɱɧɵɦ ɜɟɤɬɨɪɨɦ , ɢɦɟɟɬ ɜɢɞ
n
J
n
Jn
n
ˆˆ
J
x
n
x
J
ˆ
y
n
y
J
ˆ
z
n
z
J
ˆ
. (1)
ɋɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɷɬɨɝɨ ɨɩɟɪɚɬɨɪɚ ɟɫɬɶ
n
J
xx
Jn
yy
Jn
zz
Jn . (2)
Ⱦɥɹ ɫɨɫɬɨɹɧɢɹ
mj, ɫɩɪɚɜɟɞɥɢɜɵ ɫɨɨɬɧɨɲɟɧɢɹ
x
J 0
y
J , mJ
z
. (3)
ɍɱɢɬɵɜɚɹ ɬɚɤɠɟ ɪɚɜɟɧɫɬɜɨ T cos
z
n , ɨɤɨɧɱɚɬɟɥɶɧɨ ɡɚɩɢɲɟɦ ɢɫɤɨɦɨɟ ɫɪɟɞɧɟɟ
ɡɧɚɱɟɧɢɟ (2) ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
n
J Tcosm , (4)
ɝɞɟ – ɭɝɨɥ ɦɟɠɞɭ ɧɚɩɪɚɜɥɟɧɢɟɦ ɜɟɤɬɨɪɚ ɢ ɧɚɩɪɚɜɥɟɧɢɟɦ ɨɫɢ , ɩɨ
ɨɬɧɨɲɟɧɢɸ ɤ ɤɨɬɨɪɨɣ ɨɩɪɟɞɟɥɟɧɵ ɛɚɡɢɫɧɵɟ ɫɨɫɬɨɹɧɢɹ
T
n z
mj,.Ɉɬɦɟɬɢɦ, ɱɬɨ
ɪɚɜɟɧɫɬɜɚ (3) ɬɟɩɟɪɶ ɦɨɠɧɨ ɬɪɚɤɬɨɜɚɬɶ ɤɚɤ ɱɚɫɬɧɵɟ ɫɥɭɱɚɢ ɮɨɪɦɭɥɵ (4) ɩɪɢ
2
/
S T ɢ .0 T
7.3. ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɫɩɪɚɜɟɞɥɢɜɨ ɪɚɜɟɧɫɬɜɨ
59