Задачи по теоретической механике. Манаков Н.Л - 89 стр.

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U = a(r) +
b(θ)
r
2
H =
1
2m
Ã
p
2
r
+
p
2
θ
r
2
+
p
2
ϕ
r
2
sin
2
θ
!
+ a(r) +
b(θ)
r
2
.
S
0
1
2m
µ
∂S
0
∂r
2
+ a(r) +
1
2mr
2
"
µ
∂S
0
∂θ
2
+ 2mb(θ)
#
+
1
2mr
2
sin
2
θ
µ
∂S
0
∂ϕ
2
= E.
ϕ p
ϕ
= const
H
H = ϕ
1
(r, p
r
) = E.
ϕ
1
(r, p
r
) =
1
2m
p
2
r
+ a(r) +
1
2mr
2
ϕ
2
(θ, p
θ
).
ϕ
2
(θ, p
θ
) = p
2
θ
+ 2mb(θ) +
p
2
ϕ
sin
2
θ
= β = const.
r θ
S
0
S
0
= p
ϕ
ϕ + S
1
(r) + S
2
(θ)
§
e
a
Z a
U(r) =
e
r
3
(a · r).
S = −Et+p
ϕ
ϕ+
Z
Ã
β − 2mea cos θ −
p
2
ϕ
sin
2
θ
!
1/2
dθ+
Z
µ
2mE −
β
r
2
1/2
dr.
                                                          b(θ)
Çàäà÷à 9.5. Ïîêàçàòü, ÷òî äëÿ ïîòåíöèàëà âèäà U = a(r) +       óðàâíå-
                                                           r2
íèå Ãàìèëüòîíà-ßêîáè äîïóñêàåò ðåøåíèå ìåòîäîì ðàçäåëåíèÿ ïåðåìåííûõ
â ñôåðè÷åñêèõ êîîðäèíàòàõ.
Ðåøåíèå .  ñôåðè÷åñêèõ êîîðäèíàòàõ ôóíêöèÿ Ãàìèëüòîíà èìååò âèä (ñì.
àíàëîãè÷íóþ çàäà÷ó 8.1â)
                       Ã                   !
                                2      2
                     1        p      p ϕ             b(θ)
               H=        p2r + 2θ + 2 2      + a(r) + 2 .
                    2m        r    r sin θ            r
Óðàâíåíèå Ãàìèëüòîíà-ßêîáè (9.9) äëÿ óêîðî÷åííîãî äåéñòâèÿ S0
     µ         ¶2                   "µ          ¶2           #                µ         ¶2
 1       ∂S0                  1           ∂S0                       1             ∂S0
                    + a(r) +                         + 2mb(θ) +                              = E.
2m       ∂r                  2mr2         ∂θ                    2mr2 sin2 θ       ∂ϕ
Êîîðäèíàòà ϕ  öèêëè÷åñêàÿ, ñëåäîâàòåëüíî pϕ = const. Ïåðåïèøåì òåïåðü
âûðàæåíèå äëÿ H â âèäå
                                        H = ϕ1 (r, pr ) = E.
                                1 2               1
                        ϕ1 (r, pr ) =
                                  pr + a(r) +        ϕ2 (θ, pθ ).
                               2m              2mr2
                              2
                                              p2ϕ
               ϕ2 (θ, pθ ) = pθ + 2mb(θ) +          = β = const.
                                            sin2 θ
Òàêèì îáðàçîì, ïåðåìåííûå r è θ ðàçäåëÿþòñÿ è ñîãëàñíî (9.11), (9.12) ðåøå-
íèå óðàâíåíèÿ äëÿ S0 ìîæíî èñêàòü â âèäå
                                 S0 = pϕ ϕ + S1 (r) + S2 (θ)
(ñì. òàêæå [1, §48, ï.1] ).

Çàäà÷à 9.6. Íàïèñàòü ïîëíûé èíòåãðàë óðàâíåíèÿ Ãàìèëüòîíà-ßêîáè è
óðàâíåíèå òðàåêòîðèè â êâàäðàòóðàõ äëÿ äâèæåíèÿ çàðÿäà e â ïîëå íåïî-
äâèæíîãî ýëåêòðè÷åñêîãî äèïîëÿ ñ ìîìåíòîì a íà áîëüøèõ ðàññòîÿíèÿõ îò
äèïîëÿ.
Óêàçàíèå : Íàïðàâèòü îñü Z âäîëü âåêòîðà a è âîñïîëüçîâàòüñÿ ðåøåíèåì
      ïðåäûäóùåé çàäà÷è. Ïîòåíöèàëüíàÿ ýíåðãèÿ çàðÿäà â ýòîì ñëó÷àå ðàâ-
                 e
      íà U (r) = 3 (a · r).
                        r
Îòâåò : Ïîëíûé èíòåãðàë óðàâíåíèÿ Ãàìèëüòîíà-ßêîáè èìååò âèä:
                      Z Ã                     2
                                                  !1/2 Z µ         ¶1/2
                                             pϕ                  β
         S = −Et+pϕ ϕ+    β − 2mea cos θ −          dθ+    2mE − 2    dr.
                                           sin2 θ               r

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