Курс теоретической механики для химиков. Казаков К.А. - 69 стр.

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Φ(q, P, t) =
s
X
α=1
f
α
(q)P
α
,
f
α
(q)
p
α
=
s
X
β=1
∂f
β
(q)
∂q
α
P
β
, α = 1, ..., s ,
Q
α
=
s
X
β=1
f
β
(q)
∂P
β
∂P
α
=
s
X
β=1
f
β
(q)δ
αβ
= f
α
(q) , α = 1, ..., s ,
H
0
= H .
q → f(q),
F (x, Q, t) =
mωx
2
2
ctg Q .
p =
∂F
∂x
= mωx ctg Q , P = −
∂F
∂Q
=
mωx
2
2
1
sin
2
Q
, H
0
= H .
x =
r
2P
mω
sin Q , p =
√
2P mω sin Q .
H
0
= ωP .
˙
P = −
∂H
0
∂Q
= 0 ,
˙
Q =
∂H
0
∂P
= ω .
P = P
0
, Q = ωt + Q
0
,
P
0
, Q
0
x(t) =
r
2P
0
mω
sin(ωt + Q
0
) .
Ïðèìåð 19. Òî÷å÷íûå ïðåîáðàçîâàíèÿ. Ðàññìîòðèì êàíîíè÷åñêîå ïðåîáðàçîâàíèå, çà-
äàâàåìîå ïðîèçâîäÿùåé ôóíêöèåé
                                                s
                                                X
                                 Φ(q, P, t) =         fα (q)Pα ,                           (209)
                                                α=1

ãäå fα (q)  íåêîòîðûå ôóíêöèè. Ïî ôîðìóëàì (206)  (208) íàõîäèì
                     s
                     X ∂fβ (q)
              pα =                Pβ ,   α = 1, ..., s ,                                   (210)
                     β=1
                           ∂qα
                     Xs             X     s
                              ∂Pβ
              Qα =     fβ (q)     =     fβ (q)δαβ = fα (q) ,             α = 1, ..., s ,   (211)
                   β=1
                              ∂Pα   β=1

              H0 = H .                                                                     (212)
Óðàâíåíèå (211) ïîêàçûâàåò, ÷òî êàíîíè÷åñêèå ïðåîáðàçîâàíèÿ, ïîðîæäàåìûå ôóíêöè-
ÿìè âèäà (209) ÿâëÿþòñÿ íå ÷åì èíûì, êàê îáû÷íûìè çàìåíàìè îáîáùåííûõ êîîðäèíàò
q → f (q), ñ êîòîðûìè ìû èìåëè äåëî â ëàãðàíæåâîì ôîðìàëèçìå (èõ îáû÷íî íàçûâàþò
òî÷å÷íûìè).
Ïðèìåð 20. Ãàðìîíè÷åñêèé îñöèëëÿòîð. Ñîâåðøèì êàíîíè÷åñêîå ïðåîáðàçîâàíèå ïåðå-
ìåííûõ ëèíåéíîãî ãàðìîíè÷åñêîãî îñöèëëÿòîðà [ñì. ïðèìåð 14], çàäàâàåìîå ïðîèçâî-
äÿùåé ôóíêöèåé
                                           mωx2
                             F (x, Q, t) =      ctg Q .
                                            2
Ïî ôîðìóëàì (202)  (204) íàõîäèì
                 ∂F                                ∂F   mωx2 1
            p=      = mωx ctg Q ,        P =−         =           ,            H0 = H .
                 ∂x                                ∂Q    2 sin2 Q
Îòñþäà
                            r
                                 2P                    √
                       x=           sin Q ,     p=         2P mω sin Q .                   (213)
                                 mω
Ïîäñòàâëÿÿ ýòè âûðàæåíèÿ â ñòàðóþ ôóíêöèþ Ãàìèëüòîíà (172), ïîëó÷àåì íîâóþ
ôóíêöèþ Ãàìèëüòîíà â âèäå
                                 H 0 = ωP .
Óðàâíåíèÿ Ãàìèëüòîíà â íîâûõ ïåðåìåííûõ
                                    ∂H 0                     ∂H 0
                           Ṗ = −        = 0,         Q̇ =        = ω.
                                    ∂Q                       ∂P
Èõ ðåøåíèåì ÿâëÿåòñÿ
                                  P = P0 ,      Q = ωt + Q0 ,
ãäå P0 , Q0  íåêîòîðûå ïîñòîÿííûå. Ïîäñòàâëÿÿ åãî â (213), ïîëó÷àåì çàêîí äâèæåíèÿ
â èñõîäíûõ êîîðäèíàòàõ               r
                                       2P0
                              x(t) =       sin(ωt + Q0 ) .
                                       mω

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