Задачи по теоретической механике. Манаков Н.Л - 80 стр.

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H
0
= H +
∂F
∂t
F(q, P) =
X
α
q
α
P
α
F(q, Q) =
X
α
q
α
Q
α
F(q, P) =
X
α
f
α
(q, t)P
α
f
α
p
α
=
∂F
∂q
α
= P
α
, Q
α
=
∂F
∂P
α
= q
α
, H
0
= H
p
α
=
∂F
∂q
α
= Q
α
, P
α
= −
∂F
∂Q
α
= −q
α
, H
0
= H
Q
α
=
∂F
∂P
α
= f
α
(q, t), p
α
=
∂F
∂q
α
=
X
β
∂f
β
∂q
α
P
β
,
H
0
= H +
∂F
∂t
= H +
X
α
∂f
α
∂t
P
α
F(q, P, t) = qP + (aq −bP)t a, b
H =
p
2
2m
Q = q − bt, p = P + at, H
0
=
p
2
2m
+ aq − bP.
H
0
P Q H
0
=
(P + at)
2
2m
+ aQ− bP + abt.
˙
P = −a
˙
Q =
P + at
m
− b.
P = −at + P
0
, Q =
P
0
t
m
− bt + Q
0
.
F =
mω
2
q
2
ctg Q
                                       ∂F
      Âî âñåõ ñëó÷àÿõ H0 = H +            .
                                       ∂t
Çàäà÷à 8.19. Âûÿñíèòü ñìûñë êàíîíè÷åñêèõ ïðåîáðàçîâàíèé, çàäàâàåìûõ
ïðîèçâîäÿùèìèX ôóíêöèÿìè:
à) F(q, P) =   qα Pα ;
               α
               X
á) F(q, Q) =       qα Q α ;
               α
               X
â) F(q, P) =       fα (q, t)Pα ;
               α
fα -íåçàâèñèìûå ïðîèçâîëüíûå ôóíêöèè.
                    ∂F                 ∂F
Ðåøåíèå . à) pα =        = Pα , Qα =        = qα , H0 = H  òîæäåñòâåííîå
                   ∂qα                 ∂Pα
ïðåîáðàçîâàíèå;
         ∂F                       ∂F
á) pα =       = Qα , Pα = −           = −qα , H0 = H  ïåðåèìåíîâàíèå
         ∂qα                     ∂Qα
êîîðäèíàò â èìïóëüñû è íàîáîðîò;
         ∂F                      ∂F    X ∂fβ
â) Qα =       = fα (q, t), pα =      =         Pβ ,
         ∂Pα                     ∂qα       ∂qα
                                        β
          ∂F           X ∂fα
H0 = H +      =H+              Pα  òî÷å÷íîå ïðåîáðàçîâàíèå.
           ∂t            α
                           ∂t

Çàäà÷à 8.20. Íàéòè êàíîíè÷åñêîå ïðåîáðàçîâàíèå, ñîîòâåòñòâóþùåå ïðîèç-
âîäÿùåé ôóíêöèè F(q, P, t) = qP + (aq − bP)t, ãäå a, b  ïîñòîÿííûå. Íàïè-
ñàòü è ïðîèíòåãðèðîâàòü íîâûå óðàâíåíèÿ Ãàìèëüòîíà äëÿ ñëó÷àÿ ñâîáîä-
íîé ÷àñòèöû.
                                       p2
Ðåøåíèå . Èç (8.32), ïîëàãàÿ H =          , íàõîäèì
                                       2m
                                               p2
                                               0
                   Q = q − bt, p = P + at, H =    + aq − bP.
                                               2m
           0                   (P + at)2
                                   0
Âûðàçèì H ÷åðåç P è Q : H =               + aQ − bP + abt.
                                  2m
                                     P + at
Óðàâíåíèÿ Ãàìèëüòîíà: Ṗ = −a, Q̇ =          − b.
                                        m
                                  P0 t
Èõ ðåøåíèå: P = −at + P0 , Q =         − bt + Q0 .
                                  m
Çàäà÷à 8.21. Ïðèìåíèòü êàíîíè÷åñêîå ïðåîáðàçîâàíèå, çàäàâàåìîå ïðîèç-
                               mω 2
âîäÿùåé ôóíêöèåé F =              q ctg Q, ê ôóíêöèè Ãàìèëüòîíà îäíîìåðíîãî
                                2

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