Курс теоретической механики для химиков. Казаков К.А. - 67 стр.

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S
0
[Q(t), P (t)] =
t
2
Z
t
1
Ã
s
X
α=1
P
α
˙
Q
α
− H
0
(Q, P, t)
!
dt ,
Q
α
(t
1
) = Q
(1)
α
, Q
α
(t
2
) = Q
(2)
α
, α = 1, ..., s .
S[q(t), p(t)] S
0
[Q(t), P (t)], S[q(t), p(t)]
¯q(t), ¯p(t), S
0
[Q(t), P (t)]
¯
Q(t),
¯
P (t), ¯q(t), ¯p(t)
S[q(t), p(t)] S
0
[Q(t), P (t)]
S[q(t), p(t)] = S
0
[Q(t), P (t)] +
t
2
Z
t
1
dF (q, Q, t)
dt
dt ,
F (q, Q, t)
S[q(t), p(t)]
S
0
[Q(t), P (t)].
t
2
Z
t
1
dF (q, Q, t)
dt
dt = F (q, Q, t)|
t
2
t
1
= F (q
(2)
, Q
(2)
, t
2
) − F (q
(1)
, Q
(1)
, t
1
) .
t ∈ [t
1
, t
2
],
S S
0
,
t
1
, t
2
,
s
X
α=1
p
α
˙q
α
− H(q, p, t) =
s
X
α=1
P
α
˙
Q
α
− H
0
(Q, P, t) +
dF (q, Q, t)
dt
.
s
X
α=1
p
α
dq
α
− H(q, p, t)dt =
s
X
α=1
P
α
dQ
α
− H
0
(Q, P, t)dt + dF (q, Q, t) .
êàíîíè÷åñêèõ ïðåîáðàçîâàíèé. Êàê ìû âèäåëè, óðàâíåíèÿ Ãàìèëüòîíà (195), (196) ìî-
ãóò áûòü ïîëó÷åíû èç óñëîâèÿ ìèíèìàëüíîñòè äåéñòâèÿ

                                               Zt2 ÃX
                                                    s
                                                                                        !
                         S 0 [Q(t), P (t)] =                    Pα Q̇α − H 0 (Q, P, t) dt ,               (197)
                                               t1         α=1


ïðè óñëîâèè

                          Qα (t1 ) = Q(1)
                                      α ,           Qα (t2 ) = Q(2)
                                                                α ,            α = 1, ..., s .            (198)

Äîïóñòèì, ÷òî íàì óäàëîñü çàäàòü òàêîå ñîîòíîøåíèå ìåæäó äâóìÿ ôóíêöèîíàëàìè
S[q(t), p(t)] è S 0 [Q(t), P (t)], ÷òî åñëè S[q(t), p(t)] ïðèíèìàåò ìèíèìàëüíîå çíà÷åíèå íà
ôóíêöèÿõ q̄(t), p̄(t), òî S 0 [Q(t), P (t)] ïðèíèìàåò ìèíèìàëüíîå çíà÷åíèå íà ôóíêöèÿõ
Q̄(t), P̄ (t), ñâÿçàííûõ ñ q̄(t), p̄(t) ñîîòíîøåíèÿìè âèäà (194), è íàîáîðîò. Ïîñêîëüêó óðàâ-
íåíèÿ Ãàìèëüòîíà ïîëó÷àþòñÿ èìåííî èç óñëîâèÿ ìèíèìàëüíîñòè äåéñòâèÿ, òî ýòî îçíà-
÷àëî áû, ÷òî ïðè ïðåîáðàçîâàíèè (194) óðàâíåíèÿ (169), (170) ïåðåõîäÿò â óðàâíåíèÿ
(195), (196), ò.å. êàê ðàç êîâàðèàíòíîñòü óðàâíåíèé Ãàìèëüòîíà.
   Ñâÿæåì òåïåðü ôóíêöèîíàëû S[q(t), p(t)] è S 0 [Q(t), P (t)] ñëåäóþùèì ñîîòíîøåíèåì

                                                                       Zt2
                                               0                             dF (q, Q, t)
                          S[q(t), p(t)] = S [Q(t), P (t)] +                               dt ,            (199)
                                                                                 dt
                                                                       t1


ñ íåêîòîðîé ôóíêöèåé F (q, Q, t) ñòàðûõ è íîâûõ îáîáùåííûõ êîîðäèíàò. Ýòî ñîîòíî-
øåíèå çàäàåò æåëàåìîå ñîîòâåòñòâèå ìåæäó ìèíèìóìàìè ôóíêöèîíàëîâ S[q(t), p(t)] è
S 0 [Q(t), P (t)]. Äåéñòâèòåëüíî, âòîðîé ÷ëåí â åãî ïðàâîé ÷àñòè ìîæíî ïåðåïèñàòü òàê:

           Zt2
                 dF (q, Q, t)
                              dt = F (q, Q, t)|tt21 = F (q (2) , Q(2) , t2 ) − F (q (1) , Q(1) , t1 ) .   (200)
                     dt
           t1


 ñèëó óñëîâèé (191), (198) ïðàâàÿ ÷àñòü ïîñëåäíåãî ðàâåíñòâà ïðåäñòàâëÿåò ñîáîé
íåêîòîðóþ ôèêñèðîâàííóþ ïîñòîÿííóþ, çíà÷åíèå êîòîðîé íå çàâèñèò îò âûáîðà âèð-
òóàëüíîé òðàåêòîðèè íà ïðîìåæóòêå t ∈ [t1 , t2 ], è ïîýòîìó èç ìèíèìàëüíîñòè äåéñòâèÿ
S ñëåäóåò ìèíèìàëüíîñòü S 0 , è íàîáîðîò.
   Ðàâåíñòâî (199) áóäåò âûïîëíÿòüñÿ äëÿ âñåõ ìîìåíòîâ âðåìåíè t1 , t2 , òîëüêî åñëè
ïîäûíòåãðàëüíûå âûðàæåíèÿ â îáåèõ åãî ÷àñòÿõ òîæäåñòâåííî ñîâïàäàþò:
                 s
                 X                                 s
                                                   X                                    dF (q, Q, t)
                       pα q̇α − H(q, p, t) =              Pα Q̇α − H 0 (Q, P, t) +                   .
                 α=1                               α=1
                                                                                            dt

Ïîñëåäíåå ðàâåíñòâî ìîæåò áûòü òàêæå ïåðåïèñàíî â âèäå ñîîòíîøåíèÿ äëÿ äèôôå-
ðåíöèàëîâ
           s
           X                                        s
                                                    X
                  pα dqα − H(q, p, t)dt =                 Pα dQα − H 0 (Q, P, t)dt + dF (q, Q, t) .       (201)
           α=1                                      α=1




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