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t, φ ∂S/∂t, ∂S/∂φ.
S = S
0
(t) + S
1
(φ) + S
0
(r, θ)
dS
0
dt
= C
1
,
dS
1
dφ
= C
2
,
C
1
+
1
2m
(
µ
∂S
0
∂r
¶
2
+
C
2
2
r
2
sin
2
θ
+
1
r
2
µ
∂S
0
∂θ
¶
2
)
+
d cos θ
r
2
= 0 .
S
0
= C
1
t , S
1
= C
2
φ .
2mC
1
+
µ
∂S
0
∂r
¶
2
+
1
r
2
"
C
2
2
sin
2
θ
+
µ
∂S
0
∂θ
¶
2
+ 2md cos θ
#
= 0 .
θ
S
0
= S
3
(θ) + S
4
(r)
C
2
2
sin
2
θ
+
µ
dS
3
dθ
¶
2
+ 2md cos θ = C
3
,
2mC
1
+
µ
dS
4
dr
¶
2
+
C
3
r
2
= 0 .
S
3
=
θ
Z
θ
0
±
r
C
3
− 2md cos θ −
C
2
2
sin
2
θ
dθ ,
S
4
=
r
Z
r
0
±
r
−2mC
1
−
C
3
r
2
dr .
S = A + C
1
t + C
2
φ +
θ
Z
θ
0
±
r
C
3
− 2md cos θ −
C
2
2
sin
2
θ
dθ +
r
Z
r
0
±
r
−2mC
1
−
C
3
r
2
dr .
A,
C
1
, C
2
, C
3
, θ
0
, r
0
. θ
0
, r
0
A,
Ïåðåìåííûå t, φ âõîäÿò â ýòî óðàâíåíèå ëèøü ÷åðåç ïðîèçâîäíûå ∂S/∂t, ∂S/∂φ.  ñî-
îòâåòñòâèè ñ ìåòîäîì ðàçäåëåíèÿ ïåðåìåííûõ èùåì ðåøåíèå â âèäå
S = S0 (t) + S1 (φ) + S 0 (r, θ)
è ïðèõîäèì ê óðàâíåíèÿì
dS0 dS1
= C1 , = C2 ,
dt dφ
(µ ¶2 µ ¶2 )
1 ∂S 0 C22 1 ∂S 0 d cos θ
C1 + + 2 2 + 2 + = 0.
2m ∂r r sin θ r ∂θ r2
Ïåðâûå äâà èç ýòèõ óðàâíåíèé èíòåãðèðóþòñÿ òðèâèàëüíî:
S0 = C 1 t , S1 = C2 φ .
Ïåðåïèøåì ïîñëåäíåå óðàâíåíèå â âèäå
µ 0 ¶2 " µ 0 ¶2 #
∂S 1 C22 ∂S
2mC1 + + 2 + + 2md cos θ = 0 .
∂r r sin2 θ ∂θ
Èç ýòîé çàïèñè âèäíî, ÷òî ïåðåìåííàÿ θ îòäåëÿåòñÿ, ïîýòîìó ìû ïîëàãàåì
S 0 = S3 (θ) + S4 (r)
è ïîëó÷àåì óðàâíåíèÿ
µ ¶2
C22 dS3
+ + 2md cos θ = C3 ,
sin2 θ dθ
µ ¶2
dS4 C3
2mC1 + + 2 = 0.
dr r
Èõ ðåøåíèÿ èìåþò âèä
Zθ r
C22
S3 = ± C3 − 2md cos θ − dθ ,
sin2 θ
θ0
Zr r
C3
S4 = ± −2mC1 − dr .
r2
r0
Èòàê, ìû íàøëè ðåøåíèå óðàâíåíèÿ Ãàìèëüòîíà-ßêîáè â âèäå
Zθ r Zr r
C22 C3
S = A + C1 t + C2 φ + ± C3 − 2md cos θ − dθ + ± −2mC1 − dr . (247)
sin2 θ r2
θ0 r0
Ýòî ðåøåíèå ñîäåðæèò, ïîìèìî ïðîèçâîëüíîé àääèòèâíîé ïîñòîÿííîé A, ïðîèçâîëüíûå
ïîñòîÿííûå C1 , C2 , C3 , θ0 , r0 . Îäíàêî èçìåíåíèå θ0 , r0 ýêâèâàëåíòíî ïåðåîïðåäåëåíèþ A,
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