ISSN 1812-3368. ȼɟɫɬɧɢɤ ɆȽɌɍ ɢɦ. ɇ.ɗ. Ȼɚɭɦɚɧɚ. ɋɟɪ. «ȿɫɬɟɫɬɜɟɧɧɵɟ ɧɚɭɤɢ». 2012
53
0
M M
c
= arccos
q
r
+ arccos
0
.
q
r
ɉɨɫɬɭɩɢɜ ɜ ɨɬɧɨɲɟɧɢɢ ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɝɨ ɜɵɪɚɠɟɧɢɹ ɜ (11) ɬɚɤ
ɠɟ, ɤɚɤ ɢ ɜɵɲɟ, ɩɨɥɭɱɢɦ
0
M
M
+
0
sgn
M M
2
0
0
2
) (
)
(
qk
a
H
H
Z
0
( )
t dt
M
M
G
³
=
0
M M
c
.
(12)
ȿɫɥɢ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɫɥɭɱɚɣ ɤɪɭɝɨɜɨɝɨ ɨɞɧɨɪɨɞɧɨɝɨ ɰɢɥɢɧɞɪɢ-
ɱɟɫɤɨɝɨ ɩɨɤɪɵɬɢɹ, ɬɨ
G
(
t
)
=
G
=
const ɢ ɭɪɚɜɧɟɧɢɟ (12) ɩɪɢɜɨɞɢɬɫɹ ɤ
ɩɪɨɫɬɟɣɲɟɦɭ ɜɢɞɭ:
0
M M
0
2
0
1
1
,
a
q
H
G
M
H
ª
º
§
·
'
«
»
¨
¸
«
»
©
¹
¬
¼
0
0
M
M M
'
c
.
Ɉɛɚ ɫɩɨɫɨɛɚ ɫɜɟɞɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (8) ɤ ɩɪɨɫɬɟɣɲɟɦɭ ɜɢɞɭ ɮɨɪ-
ɦɚɥɶɧɨ ɪɚɜɧɨɩɪɚɜɧɵ. Ɍɨɬ ɮɚɤɬ, ɱɬɨ ɩɚɪɚɥɥɚɤɫ ɪɟɚɥɶɧɨ ɫɭɳɟɫɬɜɭɟɬ,
ɩɨɞɬɜɟɪɠɞɚɟɬɫɹ ɧɟɫɥɨɠɧɵɦ ɷɤɫɩɟɪɢɦɟɧɬɨɦ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɥɚɡɟɪ-
ɧɨɣ ɭɤɚɡɤɢ. ɉɪɢ ɷɬɨɦ ɨɤɚɡɵɜɚɟɬɫɹ, ɱɬɨ ɞɢɥɟɦɦɚ — ɷɮɮɟɤɬɢɜɧɵɣ ɪɚ-
ɞɢɭɫ ɢɥɢ ɷɮɮɟɤɬɢɜɧɵɣ ɭɝɨɥ — ɪɟɲɚɟɬɫɹ ɜ ɩɨɥɶɡɭ ɫɞɜɢɝɚ ɩɨ ɭɝɥɨɜɨɣ
ɤɨɨɪɞɢɧɚɬɟ: ɫɦɟɳɟɧɢɟ ɧɚɛɥɸɞɚɟɦɨɝɨ ɩɨɥɨɠɟɧɢɹ ɢɫɬɨɱɧɢɤɚ ɩɪɨɢɫɯɨ-
ɞɢɬ ɜɞɨɥɶ ɞɭɝɢ ɨɤɪɭɠɧɨɫɬɢ (
'
r
0
=
0,
'
M
0
z
0).
Ⱦɥɹ ɩɪɨɜɟɞɟɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ ɞɨɫɬɚɬɨɱɧɨ ɢɦɟɬɶ ɨɛɵɱɧɭɸ ɥɚɡɟɪ-
ɧɭɸ ɭɤɚɡɤɭ ɢ ɬɪɢ ɦɟɬɚɥɥɢɱɟɫɤɢɯ ɰɢɥɢɧɞɪɚ ɨɞɢɧɚɤɨɜɨɝɨ ɪɚɞɢɭɫɚ ɫ
ɦɟɬɚɥɥɢɱɟɫɤɨɣ ɧɢɤɟɥɢɪɨɜɚɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ. Ɉɞɢɧ ɢɡ ɰɢɥɢɧɞɪɨɜ
ɨɛɤɥɟɢɜɚɸɬ ɩɥɟɧɤɨɣ ɨɬ ɥɸɛɨɝɨ ɩɥɚɫɬɢɤɨɜɨɝɨ ɩɚɤɟɬɚ. ɍɤɚɡɤɚ ɪɚɫɩɨɥɚ-
ɝɚɟɬɫɹ ɩɚɪɚɥɥɟɥɶɧɨ ɩɨɜɟɪɯɧɨɫɬɢ ɫɬɨɥɚ, ɫɬɨɹɳɟɝɨ ɜɩɥɨɬɧɭɸ ɤ ɫɬɟɧɟ,
ɧɚ ɜɵɫɨɬɟ, ɨɛɟɫɩɟɱɢɜɚɸɳɟɣ ɨɛɥɭɱɟɧɢɟ ɜɫɟɯ ɰɢɥɢɧɞɪɨɜ, ɩɨɫɬɚɜɥɟɧ-
ɧɵɯ ɜɟɪɬɢɤɚɥɶɧɨ ɧɚ ɫɬɨɥ, ɩɪɢɦɟɪɧɨ ɧɚ ɫɟɪɟɞɢɧɟ ɢɯ ɜɵɫɨɬɵ. Ɉɞɢɧ ɢɡ
ɞɜɭɯ ɰɢɥɢɧɞɪɨɜ ɛɟɡ ɩɥɟɧɤɢ ɢɝɪɚɟɬ ɪɨɥɶ ɜɵɩɭɤɥɨɝɨ ɪɚɫɫɟɢɜɚɸɳɟɝɨ
ɡɟɪɤɚɥɚ, ɩɪɟɜɪɚɳɚɸɳɟɝɨ ɩɥɨɫɤɨɩɚɪɚɥɥɟɥɶɧɵɣ ɩɭɱɨɤ ɫɜɟɬɚ ɨɬ ɭɤɚɡɤɢ
ɜ ɪɚɫɫɟɹɧɧɵɣ ɩɭɱɨɤ, ɢɦɢɬɢɪɭɹ ɢɡɥɭɱɟɧɢɟ ɨɬ ɬɨɱɟɱɧɨɝɨ ɢɫɬɨɱɧɢɤɚ.
ɉɪɢ ɷɬɨɦ ɧɚ ɫɬɟɧɟ ɨɛɪɚɡɭɟɬɫɹ ɫɜɟɬɨɜɚɹ ɞɨɪɨɠɤɚ. ȼɫɟ ɜɡɚɢɦɧɵɟ ɪɚɫ-
ɫɬɨɹɧɢɹ (ɦɟɠɞɭ ɭɤɚɡɤɨɣ ɢ ɰɢɥɢɧɞɪɚɦɢ, ɦɟɠɞɭ ɰɢɥɢɧɞɪɚɦɢ ɢ ɫɬɟɧɨɣ)
ɦɨɠɧɨ ɜɡɹɬɶ ɩɨɪɹɞɤɚ 10 ɫɦ. ɇɚ ɩɭɬɢ ɪɚɫɫɟɹɧɧɨɝɨ ɫɜɟɬɨɜɨɝɨ ɩɭɱɤɚ
ɫɬɚɜɹɬ ɰɢɥɢɧɞɪ ɛɟɡ ɩɥɟɧɤɢ. ɇɚ ɫɜɟɬɨɜɨɣ ɞɨɪɨɠɤɟ ɩɨɹɜɥɹɟɬɫɹ ɬɟɧɟɜɚɹ
ɨɛɥɚɫɬɶ ɫ ɪɟɡɤɢɦɢ ɝɪɚɧɢɰɚɦɢ, ɤɨɬɨɪɵɟ ɫɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ ɤɚɪɚɧɞɚ-
ɲɨɦ. Ɂɚɬɟɦ ɫɬɪɨɝɨ ɧɚ ɬɨ ɠɟ ɦɟɫɬɨ, ɮɢɤɫɢɪɨɜɚɧɧɨɟ ɥɸɛɵɦ ɭɞɨɛɧɵɦ
ɫɩɨɫɨɛɨɦ, ɫɬɚɜɹɬ ɨɛɤɥɟɟɧɧɵɣ ɰɢɥɢɧɞɪ, ɢ ɬɚɤɠɟ ɨɬɦɟɱɚɸɬɫɹ ɝɪɚɧɢɰɵ
ɬɟɧɟɜɨɣ ɨɛɥɚɫɬɢ. ɇɚɛɥɸɞɚɟɬɫɹ ɢɯ ɫɦɟɳɟɧɢɟ ɜ ɨɞɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ