
Конспект лекций по дисциплине Уравнения Математической Физики (УМФ)
.pdf§4 nELINEJNYE u~p PERWOGO PORQDKA. uRAWNENIE gAMILXTONA–qKOBI
4.1oSNOWNYE OPREDELENIQ I POSTANOWKA ZADA^I kO[I DLQ NESTACIONARNOGO URAWNENIQ gAMILXTONA–qKOBI
oB]IJ WID NELINEJNOGO u~p PERWOGO PORQDKA :
|
|
|
∂u |
|
|
(4.1) |
L |
x, |
∂x |
, u |
= 0, |
GDE x = (x1, . . . , xn) Rxn, |
∂u |
= |
∂u |
, . . . , |
∂u |
, A L(x, p, u) – GLADKAQ WE]ESTWENNAQ |
|
|
|
||||
∂x |
∂x1 |
∂xn |
FUNKCIQ WSEH SWOIH (2n + 1) ARGUMENTOW. ~ASTNYJ SLU^AJ \TOGO URAWNENIQ
(4.2) |
L |
x, |
∂u |
= 0 |
|
|
|
|
|
|
||
|
|
|
|
|
|
|
||||||
∂x |
|
|
|
|
|
|
||||||
NAZYWA@T URAWNENIEM gAMILXTONA |
– |
qKOBI |
, |
A FUNKCI@ |
L(x, p) |
PEREMENNYH |
x |
|
n |
|||
n |
|
|
|
|
|
Rx |
||||||
I p Rp – FUNKCIEJ gAMILXTONA ILI GAMILXTONIANOM. |
|
|
|
n+1 |
, |
|||||||
rASSMOTRIM ^ASTNYJ SLU^AJ URAWNENIQ gAMILXTONA–qKOBI W PROSTRANSTWE Rx |
|
KOGDA L QWLQETSQ LINEJNOJ FUNKCIEJ PO ODNOJ IZ SWOIH PEREMENNYH p Rnp+1, NAPRIMER, PO KOORDINATE pn+1. oBOZNA^IM SOPRQVENNU@ pn+1 PEREMENNU@ xn+1 ^EREZ t (xn+1 = t), S^ITAQ, ^TO t – FIZI^ESKOE WREMQ. tOGDA URAWNENIE (4.2) PRIMET WID
(4.3) |
L = |
∂u |
+ H x1, x2, . . . , xn, t, |
∂u |
, |
∂u |
, . . . , |
∂u |
= 0. |
|
|
|
|
||||||
∂t |
∂x1 |
∂x2 |
∂xn |
w KLASSI^ESKOJ MEHANIKE PRINQTO OBOZNA^ATX W \TOM SLU^AE NEIZWESTNU@ FUNKCI@ u(x, t) ^EREZ S(x, t) I NAZYWATX EE DEJSTWIEM [ ], A SOOTWETSTWU@]EE URAWNENIE (4.3) – NE-
STACIONARNYM URAWNENIEM gAMILXTONA–qKOBI. tAKIM OBRAZOM, NESTACIONARNOE URAW-
NENIE gAMILXTONA–qKOBI OTNOSITELXNO FUNKCII (n + 1) PEREMENNOJ S(x, t), GDE t R – WREMQ, A x = (x1, x2, . . . , xn) – TO^KA KONFIGURACIONNOGO PROSTRANSTWA
W WIDE
(4.4) |
∂S |
+ H |
x, t, |
∂S |
= 0. |
|
|
||||
∂t |
∂x |
zDESX H(x, t, p) – GLADKAQ FUNKCIQ WSEH SWOIH PEREMENNYH, KOTORAQ TAKVE NAZYWAETSQ
FUNKCIEJ gAMILXTONA.
dLQ URAWNENIQ (4.4) POSTAWIM ZADA^U kO[I S NA^ALXNYM USLOWIEM
(4.5) |
S|t=0 = S0(x), |
x Ω0 Rxn, |
GDE S0(x) C∞(Ω0) – GLADKAQ FUNKCIQ. |
|
nIVE IZLOVIM ALGORITM RE[ENIQ \TOJ ZADA^I, A TAKVE POSTANOWKU I ALGORITM RE- [ENIQ ZADA^I kO[I DLQ SLU^AQ OB]EGO (STACIONARNOGO) URAWNENIQ gAMILXTONA–qKO-
BI (4.2).
31

4.2aLGORITM a3 RE[ENIQ ZADA^I kO[I
DLQ NESTACIONARNOGO URAWNENIQ gAMILXTONA–qKOBI
rASSMOTRIM ZADA^U (4.4)–(4.5).
1◦. wYPISATX HARAKTERISTI^ESKU@ SISTEMU DLQ (4.4) – SISTEMU gAMILXTONA W 2n-MERNOM
FAZOWOM PROSTRANSTWE R2n = Rnp × Rnx, GDE Rnx – KONFIGURACIONNOE PROSTRANSTWO, Rnp – IMPULXSNOE PROSTRANSTWO:
x˙ = pH(x, t, p),
(4.6)
p˙ = − xH(x, t, p).
fUNKCIQ gAMILXTONA (KLASSI^ESKIJ GAMILXTONIAN) H(x, t, p) OPREDELQETSQ PO WI-
DU (4.4).
2◦. pOSTAWITX DLQ (4.6) ZADA^U kO[I:
(4.7) |
|
|
|
x |
x |
, |
S |
x |
Ω0 |
, |
|
S x |
|
|
S x , |
|
|
|
|
|
p|t=0 = |
|
0 |
|
x 0 |
|
|
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
= ( x 0)( 0) = ( 0)x( 0) |
|
|||||||
|
|
|
|
|t=0 = x 0( ) |
x=x0 |
|
|||||||||||
I NAJTI n- |
|
|
|
|
(x0 – |
n-MERNYJ PARAMETR) |
SEMEJSTWO RE[ENIJ ZADA- |
||||||||||
|
|
|
|
PARAMETRI^ESKOE |
|
|
|
|
|
|
|
|
|
|
|||
^I (4.6)–(4.7) NA OTREZKE [0, T ], |
T > 0: |
|
|
|
|
|
|
|
|||||||||
|
|
|
|
|
|
|
|
|
x = X(x0, t), |
|
|
|
|
|
|||
(4.8) |
|
|
|
|
Lx0 : p = P (x0, t), |
0 ≤ t ≤ T. |
|
||||||||||
Lx0 R |
2n |
– HARAKTERISTIKA, ILI FAZOWAQ |
|
|
n , |
STARTU@]AQ IZ TO^KI |
(x0, p0), |
||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
TRAEKTORIQ |
|
||||
GDE |
p0 |
= |
|
S0(x0) (RIS. |
4.1). pROEKCIQ Lx0 |
NA Rx ESTX LU^ , ILI TRAEKTORIQ |
|||||||||||
x = |
X(x0, t), 0 ≤ t ≤ T , KLASSI^ESKOJ ^ASTICY, STARTU@]AQ IZ TO^KI x0 S NA- |
||||||||||||||||
^ALXNYM IMPULXSOM p0 = S0(x0). |
|
|
|
|
|
|
|
|
lU^ x = X(x0,t) NA OTREZKE [0,T] ( t x = X(x0,t) — KOORDINATY LU^A ).
32

3◦. wY^ISLITX DEJSTWIE NA HARAKTERISTIKE Lx0 :
|
|
0 |
t |
|
|
|
|
|
|
|
|
||
(4.9) |
˜ |
, t) = S0(x0) + |
p, pH |
H |
|
x=X(x ,τ), dτ. |
S(x0 |
|
|||||
|
|
. |
− |
|
p=P (x00,τ) |
|
|
|
|
|
4◦. rAZRE[ITX PERWYE n URAWNENIJ SISTEMY (4.8) OTNOSITELXNO PARAMETRA x0, PREDPO-
LAGAQ, ^TO QKOBIAN |
|
|
|
|
|
|
(4.10) |
Jx(x0, t) = |
DX(x0, t) |
= 0 |
(x0 Ω0, 0 ≤ t ≤ T ): |
||
|
|
|||||
Dx0 |
||||||
|
|
x = X(x0, t) x0 = x0(x, t). |
||||
5◦. pOSTROITX FUNKCI@ |
|
|
S˜(x0 |
|
x0=x0(x,t). |
|
(4.11) |
|
S(x, t) = |
, t) |
|||
|
|
|
|
) |
|
* |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
tEOREMA 4.1. pUSTX FUNKCIQ H(x, t, p) DWAVDY NEPRERYWNO DIFFERENCIRUEMA PO WSEM
SWOIM ARGUMENTAM, S0(x) C(2) |
Rxn |
I PUSTX |
pH(x, t, p) |
= |
0. tOGDA FORMULA (4.11) |
||||||||||
OPREDELQET EDINSTWENNOE |
DIFFERENCIRUEMOE PO |
x I t RE[ENIE |
ZADA^I kO[I (4.4)–(4.5) |
||||||||||||
|
|
|
|
|
|||||||||||
W ”POLOSE” Πx,t = (x, t) |
|
Rn+1: x |
= X(x |
, t), x |
0 |
Ω , |
t |
|
|
[0, T ], |
Jx(x |
, t) = 0 |
|
||
(RIS. 4.2). |
|
|
0 |
|
|
0 |
|
|
|
0 |
|
dOKAZATELXSTWO BUDET PRIWEDENO NIVE (SM. §5).
pRIMER 4.1. sLEDUQ PUNKTAM ALGORITMA a3, NAJDEM RE[ENIE ZADA^I kO[I
|
∂S |
|
1 |
|
2 |
αx2 |
|
|
|
|
||||
|
|
|
|
∂S |
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
R1 |
|
|
∂t |
+ 2m ∂x |
= 0, |
|
x R1, t > 0, m > 0, |
||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
= S0(x) = , |
α |
|
|
. |
||||||
S(x, t) |
t=0 |
|
|
|||||||||||
|
|
|
|
|
|
2 |
|
|
|
|
33

1) iMEEM DLQ H(x, t, p) = |
|
p2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
SISTEMU gAMILXTONA |
|
|
|
|
|
|
|||||||
|
2m |
|
|
|
|
|
|
||||||||
|
|
|
|
|
∂ |
|
p |
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
−∂x |
|
|
|
|
|
|
|
|||||
|
|
|
x˙ = |
|
∂pH(x, t, p) = m, |
|
|
|
|||||||
KOTORAQ \KWIWALENTNA |
|
|
|
|
|
∂ |
|
mx |
F |
|
|
|
|
|
|
|
|
|
p˙ = |
|
|
H(x, t, p) = 0, |
|
|
|
||||||
|
|
|
|
|
|
|
|
|
|||||||
|
URAWNENI@ nX@TONA |
¨ = |
|
x = 0. |
|
||||||||||
|
|
|
|
|
|
|
|
|
|
|
|||||
2) nA^ALXNYE DANNYE DLQ NEE: |
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
x t=0 = x0, |
|
x0 |
|
R1 |
, |
||||||||
|
|
p|t=0 = S0 |
(x0) = αx0, |
|
|||||||||||
|
| |
|
|
|
|
|
|
|
|
|
|
|
A ZNA^ENIE FUNKCII gAMILXTONA NA FAZOWOJ TRAEKTORII Lx0
|
|
p2 |
|
|
α2x |
2 |
|
H|Lx0 |
= |
2m |
Lx0 |
= |
2m0 |
|
= EKIN |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
ESTX KINETI^ESKAQ \NERGIQ ^ASTICY MASSOJ m, DWIVU]EJSQ RAWNOMERNO I PRQMOLI- NEJNO:
αx0
x = x0 + m t = X(x0, t), p = αx0 = const.
3) wY^ISLIM |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
p |
∂H |
− H = p |
|
p |
− |
1 |
p2 = |
|
p2 |
|
|
|
||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
, |
|
|
|
|||||||||||
|
∂p |
m |
2m |
2m |
|
|
|
||||||||||||||||||||
|
|
αx 2 |
0 |
t |
|
p2 |
|
|
|
|
|
|
|
|
|
αx 2 |
|
2x 2 |
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
α |
|||||||||||||||
|
S˜(x0, t) = |
20 + . |
|
|
2m p=αx0 dt = |
20 + |
2 0 t, |
||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
αx0 |
|
|
|
|
|
|
|
|
|
αt |
|
|
|
||||||
|
|
x = x0 + |
|
|
|
t = x0 1 + |
|
, |
|
|
|
||||||||||||||||
|
|
|
m |
|
m |
|
|
|
|||||||||||||||||||
|
|
|
x0 = |
|
|
x |
|
|
|
|
|
= x0(x, t), |
|
|
|
|
|
|
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||
|
|
1 + |
αt |
|
|
|
|
|
|
||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
|
|
|
|
|
|
|
m |
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||
S^ITAQ, ^TO |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
αt |
|
|
|
|
|
|
|
|
|
|
|
∂X |
|
|
|
|
|
|
||||||
(4.12) |
1 + |
|
|
= Jx |
(x0, t) = |
|
|
(x0, t) = 0. |
|
|
|
||||||||||||||||
m |
∂x0 |
|
|
|
4)tOGDA RE[ENIE ISHODNOJ ZADA^I kO[I ZADAETSQ (PRI WYPOLNENII USLOWIQ (4.12)) FORMULOJ
S(x, t) = |
' |
|
x02 |
1 + |
|
|
( |
|
|
≡ |
|
|
|
|
1 + |
|
= |
|
|
|
|
. |
2 |
m |
= |
x |
|
αt |
|
2 |
m |
|
|
αt |
|||||||||||
|
|
|
x0 |
|
|
αx2 |
|
|
|
αt |
|
|
αx2 |
|||||||||
|
|
α |
|
αt |
|
|
m |
|
|
|
|
|
|
|||||||||
|
|
|
|
|
|
|
|
|
|
m |
|
|
|
|
|
|
m |
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1+ αt |
|
2 1 + |
|
|
|
|
|
|
2 |
1 + |
|
|
|
34

|
2 |
|
|
SU]ESTWUET GLADKOE RE[ENIE ZADA^I |
|
||||||
tAKIM OBRAZOM, POLOSKA Πx,t Rx,t, W KOTOROJ |
, |
||||||||||
|
1 |
|
ESLI |
|
|
||||||
ZAWISIT OT ZNA^ENIJ PARAMETRA α: |
ESLI |
α ≥ 0, |
TO |
Πx,t = Rx |
× [0, +∞), |
α < 0, |
TO |
||||
m |
|
|
|
|
|
Πx,t = R1x × [0, T ], GDE T < t , t = − α .
zAME^ANIE. eSLI RASSMOTRETX BOLEE OB]EE URAWNENIE, ^EM (4.4), WIDA
∂S
∂t + H(x, t, xS, S) = 0,
TO W \TOM SLU^AE ALGORITM RE[ENIQ ZADA^I kO[I ANALOGI^EN. pRI \TOM HARAKTERISTI- ^ESKAQ SISTEMA DLQ \TOGO URAWNENIQ W PROSTRANSTWE R2n+1 IMEET WID
x˙ = pH, |
∂H |
, |
p˙ = −xH − p |
∂S |
|
|
|
|
˙ |
|
|
S = p, pH − H(x, t, p, S).
4.3zADA^A kO[I DLQ STACIONARNOGO URAWNENIQ gAMILXTONA–qKOBI I ALGORITM a4 EE RE[ENIQ
rASSMOTRIM STACIONARNOE URAWNENIE gAMILXTONA–qKOBI
|
|
∂S |
|
(4.13) |
H x, |
∂x |
= 0, x Rn. |
zADA^EJ kO[I DLQ URAWNENIQ (4.13) NAZYWAETSQ SLEDU@]AQ ZADA^A: NAJTI RE[ENIE URAW- NENIQ, UDOWLETWORQ@]EE NA GLADKOJ GIPERPOWERHNOSTI
|
x: x = X0(ξ), ξ = (ξ1, . . . , ξn−1) D, rank |
∂X0 |
|
(ξ) = n − 1/ |
||
γn−1 = |
i |
n |
|
|||
∂ξj |
(n 1) |
|||||
|
|
|
× |
− |
|
|
DANNYM |
|
|
|
|
|
ξ |
W Rxn, GDE X0(ξ) – ZADANNAQ GLADKAQ WEKTOR-FUNKCIQ, D – OBLASTX W Rn−1, NA^ALXNYM |
||||||
(4.14) |
|
S|γn−1 = S0(ξ), |
|
|
|
|
(4.15) |
|
S|γn−1 = P 0(ξ), |
|
|
|
|
GDE S0 I P 0 – ZADANNYE GLADKIE FUNKCIQ I WEKTOR-FUNKCIQ, POD^INENNYE USLOWIQM |
||||||
(4.16) |
|
|
|
|
|
|
|
(SOGLASOWANIQ γn−1 I P 0(ξ) S URAWNENIEM (4.13)); |
|||||
H X0(ξ), P 0(ξ) = 0 |
||||||
(4.17) |
|
|
|
|
|
|
|
n |
|
|
|
|
|
|
|
|
|
|
|
|
dS0(ξ) = |
P 0i(ξ) dX0i(ξ) |
(SOGLASOWANIQ P 0 S DIFFERENCIALOM FUNKCII S0). |
i=1
35
aLGORITM a4 RE[ENIQ ZADA^I kO[I (4.13)–(4.17)
1◦. wYPISATX HARAKTERISTI^ESKU@ SISTEMU DLQ URAWNENIQ (4.13) — SISTEMU gAMILXTO- NA S GAMILXTONIANOM H(x, p):
(4.18)
x˙ = pH(x, p), p˙ = −xH(x, p),
GDE (x, p) R2n, x˙ = |
dx |
|
dp |
|
|
|
|
|
|
||
|
, p˙ = |
|
, I NAJTI (n−1)-PARAMETRI^ESKOE (ξ – (n−1)-MERNYJ |
||||||||
dτ |
dτ |
||||||||||
PARAMETR) SEMEJSTWO EE RE[ENIJ: |
|
|
|
|
|
|
|||||
|
|
|
|
|
x = X ξ, τ |
, |
|
|
|||
(4.19) |
|
|
Lξ: p = P ((ξ, τ)) |
|
(|τ |
| < τ0) |
|||||
S NA^ALXNYMI DANNYMI |
|
|
|
|
|
|
|||||
(4.20) |
|
|
|
|
x τ=0 = X0(ξ), |
|
|
||||
|
|
|
|
p|τ=0 = P 0(ξ). |
|
|
|||||
|
|
|
| |
|
|
|
|
|
|
||
2◦. wY^ISLITX DEJSTWIE NA HARAKTERISTIKE Lξ: |
|
|
|||||||||
|
|
|
|
|
τ |
|
|
|
|
|
|
|
|
|
0 |
|
|
|
|
||||
(4.21) |
S˜(ξ, τ) = S0(ξ) + |
p, |
|
|
pH(x, p) |
|
x=X(ξ,τ , dτ . |
||||
|
|
|
. |
|
|
|
|
p=P (ξ,τ )) |
|||
|
|
|
|
|
|
3◦. rAZRE[ITX PERWYE n URAWNENIJ SISTEMY (4.19) OTNOSITELXNO τ I ξ Rn−1 W PREDPOLOVENII, ^TO WYPOLNENO USLOWIE
(4.22)
(4.23)
4◦. pOSTROITX FUNKCI@
(4.24)
J |
ξ, τ |
) = |
|
DX(ξ, τ) |
, |
D, τ |
| |
< τ |
: |
|
|
D(ξ, τ) |
|||||||||
( |
|
|
= 0 ξ |
| |
0 |
|||||
|
|
x = X(ξ, τ) |
ξ = ξ(x), |
|
|
|
||||
|
|
τ = τ(x). |
|
|
|
|||||
|
|
|
S(x) = S˜(ξ, τ) |
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|||
|
|
|
ξ=ξ(x) . |
|
|
|
|
|||
|
|
|
) |
* |
|
|
|
|
|
|
|
|
|
τ=τ(x) |
|
|
|
|
tEOREMA 4.2. pUSTX WYPOLNENO USLOWIE ”TRANSWERSALXNOSTI” |
|
||||||
|
|
∂X0 |
|
|
|
|
|
(4.25) |
det |
i |
|
, pH X0(ξ), P 0(ξ) = 0. |
|
||
∂ξj |
|
|
|||||
tOGDA |
FORMULA (4.24) OPREDELQET |
EDINSTWENNOE |
GLADKOE RE[ENIE |
ZADA^I kO[I |
|||
(4.13)–(4.15) W OKRESTNOSTI V (γ) = |
|
|
|
, J(ξ, τ) = 01. |
|||
0x Rn: x = X(ξ, τ), ξ D, |τ| < τ0 |
36
dOKAZATELXSTWO \TOJ TEOREMY PO^TI DOSLOWNO POWTORQET DOKAZATELXSTWO TEOREMY 4.1.
uPRAVNENIE. dOKAVITE TEOREMU 4.2.
pRIMER 4.2. dLQ URAWNENIQ \JKONALA W GEOMETRI^ESKOJ OPTIKE (SM. TAKVE PUNKT ??).
|
|
|
|
∂S |
2 |
|
|
∂S |
|
|
2 |
|
|
|
|
|
|
|
|||
(4.13 ) |
|
|
|
|
+ |
|
|
= 1, |
(x1, x2) R2, |
||||||||||||
|
|
∂x1 |
∂x2 |
||||||||||||||||||
RE[IM ZADA^U kO[I S NA^ALXNYMI USLOWIQMI |
|
|
|
|
|||||||||||||||||
(4.14 ) |
|
|
|
|
|
|
|
|
S|γ = S0 = 0, |
|
|
|
|||||||||
(4.15 ) |
|
|
|
|
|
|
|
|
|
S |
| |
γ = P 0(ξ), |
|
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
GDE γ – OKRUVNOSTX x12 + x22 = 1. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
wWEDEM ESTESTWENNU@ PARAMETRIZACI@ KRIWOJ γ: |
|
|
|
|
|||||||||||||||||
x1 = X01(ξ) = cos ξ, |
|
|
|
x2 = X02(ξ) = sin ξ, |
ξ [0, 2π]. |
||||||||||||||||
nAJDEM WEKTOR |
|
0 |
(ξ) = P |
0 |
|
0 |
|
|
|
|
|
|
|
USLOWIJ SOGLASOWANIQ |
|||||||
(4.16 ) |
P |
|
1 |
(0ξ), P0 2(ξ) IZ0 |
|
2 |
+ P |
0 |
2 |
− 1 = 0, |
|||||||||||
|
|
H P 1, P |
2 |
= P |
1 |
|
2 |
|
|||||||||||||
0 = P 01(ξ) dX01(ξ) + P 02(ξ) dX0 |
2(ξ) = P 01(ξ) d(cos ξ) + P 02(ξ) d(sin ξ) = |
||||||||||||||||||||
(4.17 ) |
−P 01(ξ) sin ξ dξ + P 02(ξ) cos ξ dξ. |
|
|
|
|
||||||||||||||||
= |
|
|
|
|
|||||||||||||||||
rE[ENIE POSLEDNEJ SISTEMY: |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
|
|
|
|
|
|
|
|
P±0 |
1(ξ) = ± cos ξ, |
|
|
|
|||||||||
|
|
|
|
|
|
|
|
P±0 |
2(ξ) = ± sin ξ. |
|
|
|
|||||||||
oTS@DA USLOWIE (?? ) PRINIMAET WID |
|
|
|
|
|
|
± |
|
|
|
|
||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
cos ξ |
|
|
|
|
(4.15 ) |
|
|
|
|
|
|
S± |
γ = |
± sin ξ . |
|
|
||||||||||
|
|
|
|
|
|
|
|
|
tAKIM OBRAZOM, W ZAWISIMOSTI OT WYBORA ZNAKA ( + ILI − ) MY POLU^AEM ODNU IZ DWUH ZADA^ kO[I WIDA (?? )–(?? ). rE[ENIQ \TIH ZADA^ BUDEM OBOZNA^ATX ^EREZ S±(x). sLEDUQ PUNKTAM ALGORITMA a4, NAJDEM S±(x).
1)gAMILXTONIAN DLQ URAWNENIQ (?? ) IMEET WID H(x, p) = p12 + p22 − 1. wYPISYWAEM SISTEMU gAMILXTONA:
x˙ = 2p ,
1 1
x˙ 2 = 2p2,
p˙1 = 0,
p˙2 = 0.
37

2) nA^ALXNYE USLOWIQ DLQ NEE:
|
x1 τ=0 = cos ξ, |
|||
| |
|
|
|
|
|
| |
|
± |
|
|
|
|
||
x2|τ=0 |
= sin ξ, |
|||
p1± τ=0 = |
|
cos ξ, |
||
|
|
|
|
|
|
|
|
|
|
|
|
|
± sin ξ. |
|
p2±|τ=0 |
= |
nAHODIM RE[ENIE POLU^ENNOJ ZADA^I kO[I:
|
x1± |
(4.19 ) |
x2± |
|
|
|
p1± |
|
|
|
|
p±2
=cos ξ ± 2τ cos ξ = cos ξ · (1 ± 2τ),
=sin ξ ± 2τ sin ξ = sin ξ · (1 ± 2τ),
=± cos ξ,
=± sin ξ.
3) wY^ISLIM
τ |
|
|
|
|
|
|
|
|
0 |
|
|
|
|
|
τ |
|
|
τ |
(2p 2 |
|
|
p1= cos ξ, dτ = |
|
|
||
S˜±(ξ, τ) = |
+ 2p 2) |
|
|
|
||||
. |
1 |
2 |
|
p2=±± sin ξ |
|
|
|
|
0 |
|
|
|
|
= |
0 |
2 · 1 dτ = 2τ. |
|
= . |
2 (± cos ξ)2 + (± sin ξ)2 |
dτ |
. |
4)rAZRE[IM PERWYE DWA URAWNENIQ SISTEMY (?? ) OTNOSITELXNO τ. wOZWEDQ W KWADRAT I SLOVIW OBE ^ASTI \TIH URAWNENIJ, NAHODIM
|
τ± = |
√ |
|
− 1 |
, |
|
|
|
|
|
|
|||
|
x12 + x22 |
|
|
|
|
|
||||||||
|
|
|
|
|
|
|
|
|
|
|||||
|
1 |
|
√ |
2±2 |
|
|
|
|
|
|
|
|
|
|
|
|
− |
2 |
− 1 |
|
|
|
|
|
|
||||
|
τ2± = |
|
x1 + x2 |
. |
|
|
|
|
|
|||||
|
|
±2 |
|
|
|
|
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
pRI \TOM QKOBIAN IMEET WID |
|
|
|
|
|
|
|
|
|
|
|
|
||
J± = det |
− sin ξ · (1 ± 2τ) |
±2 cos ξ |
= |
|
2 |
− |
4τ. |
|||||||
|
cos ξ · (1 ± 2τ) |
±2 sin ξ |
|
|
|
5) o^EWIDNO, ^TO WTOROJ IZ \TIH KORNEJ NE UDOWLETWORQET NA^ALXNOMU USLOWI@. tOGDA
,
S±(x) = ± x12 + x22 1.
zAME^ANIE. zADA^A kO[I (4.4)–(4.5) DLQ NESTACIONARNOGO URAWNENIQ gAMILXTONA– qKOBI (4.4) I ALGORITM a3 EE RE[ENIQ QWLQ@TSQ ^ASTNYMI SLU^AQMI ZADA^I kO[I DLQ OB]EGO (STACIONARNOGO) URAWNENIQ gAMILXTONA–qKOBI (4.13) I ALGORITMA a4 SOOTWETSTWENNO. dEJSTWITELXNO, POLNYJ GAMILXTONIAN H, OTWE^A@]IJ URAWNENI@ (4.4), ESTX
38
FUNKCIQ, OPREDELENNAQ NA RAS[IRENNOM FAZOWOM PROSTRANSTWE R2n+2 = R2x,pn × R2pt,t,
WIDA
(4.26) |
|
|
H(x, t, p, pt) = pt |
+ H(x, t, p), |
|||||
|
|
|
|
|
∂S |
∂S |
|
||
I \TO URAWNENIE (4.4) PRINIMAET WID H x, t, |
∂x |
, |
∂t |
= 0. sOOTWETSTWU@]AQ SISTEMA |
|||||
gAMILXTONA W R2n+2 IMEET WID |
|
|
|
|
|
|
|||
(4.27) |
x˙ = pH(x, t, p) = pH, |
|
|
p˙ = −xH(x, t, p) = −xH, |
|||||
|
|
∂ |
∂ |
|
|
t˙ = 1 |
|||
(4.28) |
p˙t = − |
|
H(x, t, p) = − |
|
H, |
|
|
||
∂t |
∂t |
|
|
||||||
(ZDESX TO^KA OZNA^AET DIFFERENCIROWANIE PO PARAMETRU τ, NAPRIMER, p˙t = dpt ). |
|||||||||
|
|
|
|
|
|
|
|
|
dτ |
pROEKCI@ FAZOWOJ TRAEKTORII SISTEMY (4.27)–(4.28) NA KONFIGURACIONNOE
PROSTRANSTWO Rn+1 |
|
x,t |
|
lx,t(x0) = (x, t): x = X(x0, τ), t = τ |
(x0 Ω0, τ R+1 ) |
NAZYWA@T PROSTRANSTWENNO-WREMENNYM LU^OM W PROSTRANSTWE-WREMENI Rnx,t+1. iNTEGRIROWANIE SISTEMY gAMILXTONA (4.27)–(4.28), O^EWIDNO, \KWIWALENTNO (ESLI
S^ITATX, ^TO τ = t) INTEGRIROWANI@ UKORO^ENNOJ SISTEMY gAMILXTONA W 2n-MERNOM
FAZOWOM PROSTRANSTWE R2x,pn H(x, t, p), KOTORAQ I ISPOLXZOWALASX W AL- GORITME a3.
§5 oBOSNOWANIE ALGORITMOW RE[ENIQ ZADA^ kO[I DLQ URAWNENIQ gAMILXTONA–qKOBI
dOKAZATELXSTWO TEOREM 4.1 I 4.2 OSNOWYWAETSQ NA LEMME gAMILXTONA.
lEMMA gAMILXTONA (NESTACIONARNYJ WARIANT). iMPULXS NA FAZOWOJ TRAEKTO-
RII RAWEN GRADIENTU DEJSTWIQ NA LU^E. |
|
|
|
|
|
|
|
|
|
|
|||||||
iNYMI SLOWAMI, W TO^KE x = X(x0, t), QWLQ@]EJSQ PROEKCIEJ TO^KI |
X(x0, t), P (x0, t) |
||||||||||||||||
FAZOWOJ TRAEKTORII L |
NA KONFIGURACIONNOE PROSTRANSTWO |
( |
RIS |
. |
4.1), W L@BOJ MOMENT |
||||||||||||
|
|
|
|
|
x0 |
|
|
|
|
|
|
|
|
|
|
||
WREMENI t [0, T ] PRI WYPOLNENII USLOWIQ (4.10) IMEET MESTO RAWENSTWO |
|||||||||||||||||
(5.1) |
|
|
|
|
P (x0, t) = xS |
X(x0, t), t , |
|
|
|
|
|
|
|||||
GDE FUNKCIQ |
S(x, t) |
OPREDELENA FORMULAMI |
|
|
|
|
|
|
|
|
|
|
|||||
|
|
|
|
|
|
(4.6)–(4.11). |
|
|
|
|
|
|
|
||||
dOKAZATELXSTWO. pROWEDEM DOKAZATELXSTWO DLQ SLU^AQ n = 1 (x R1) (DLQ SLU^AQ |
|||||||||||||||||
n |
≥ 2 |
DOKAZATELXSTWO ANALOGI^NO). |
|
|
|
|
|
|
|
|
|
|
|
||||
|
|
|
|
ALGORITMA a |
3 IMEEM |
|
|
|
|
|
|
|
|
|
|
||
w SILU PUNKTOW 3◦–5◦ |
|
|
|
|
|
|
|
|
|
|
|
|
|||||
|
|
|
|
|
|
|
|
|
t |
|
|
|
|
|
|
|
|
(5.2) |
S X(x0, t), t |
= S˜(x0, t) = S0(x0) + |
p, |
pH |
− |
H |
x=X(x ,t ), dt , |
||||||||||
|
|
|
|
|
|
|
|
. |
|
|
|
|
p=P (x00,t ) |
||||
|
|
|
|
|
|
0 |
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
39

GDE SKALQRNOE PROIZWEDENIE
∂H
p, pH = p ∂p (x, t, p).
pRODIFFERENCIRUEM DANNOE RAWENSTWO (5.2) PO PARAMETRU x0, ISPOLXZUQ PRI \TOM SIS- TEMU gAMILXTONA.
|
∂S |
X(x0, t), t |
∂X |
(x0, t) = |
|
∂ |
S0(x0) + |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||||||||||||||||
|
∂x |
|
|
t |
|
|
|
|
|
∂x0 |
|
|
|
|
|
|
|
∂x0 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
, t ) dt = |
|||||||||||||||||
|
|
+ .0 |
|
|
∂P |
|
|
|
|
|
|
∂H |
|
|
|
|
|
∂ ∂H |
∂H ∂X |
|
|
|
|
|
|
∂H ∂P |
|
|||||||||||||||||||||||||||
|
|
|
|
|
|
|
(x0, t ) |
|
|
+ P |
|
|
|
|
|
|
|
− |
|
|
|
|
|
(x0, t ) − |
|
|
|
(x0 |
||||||||||||||||||||||||||
|
|
|
∂x0 |
∂p |
∂x0 |
∂p |
∂x |
∂x0 |
∂p |
∂x0 |
||||||||||||||||||||||||||||||||||||||||||||
|
|
|
|
∂ |
|
|
|
|
|
|
|
|
t |
|
|
|
|
∂ |
|
|
|
|
|
|
|
∂X |
|
|
||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||||||||||||||||||
|
|
= |
|
S0 |
(x0) + |
.0 |
|
P |
|
|
|
X˙ (x0, t ) + P˙ (x0, t ) |
|
|
|
(x0, t ) dt = |
|
|||||||||||||||||||||||||||||||||||||
|
|
∂x0 |
|
∂x0 |
∂x0 |
|
||||||||||||||||||||||||||||||||||||||||||||||||
|
|
|
|
∂ |
|
|
|
|
|
|
|
|
t |
|
∂ |
P (x0, t ) |
∂X |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||
|
|
= |
|
S0 |
(x0) + |
.0 |
|
|
|
|
(x0, t ) dt = |
|
|
|||||||||||||||||||||||||||||||||||||||||
|
|
∂x0 |
|
|
∂t |
∂x0 |
|
|
||||||||||||||||||||||||||||||||||||||||||||||
|
|
|
|
∂ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
∂X |
|
|
|
|
|
|
∂X |
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||
|
|
= |
|
S0 |
(x0) + P (x0, t) |
|
|
(x0, t) − P (x0, 0) |
|
|
(x0, 0). |
|
|
|||||||||||||||||||||||||||||||||||||||||
|
|
∂x0 |
∂x0 |
∂x0 |
|
|
||||||||||||||||||||||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
∂X |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
∂S0 |
|
|
|
∂ |
|
|
||||||||||||||||||
w SILU TOGO, ^TO |
|
(x0, 0) = 1 I P (x0, 0) = |
|
|
(x0) = |
|
|
|
|
S0(x0) (SM. FORMULU (4.7)), |
||||||||||||||||||||||||||||||||||||||||||||
∂x0 |
∂x |
∂x0 |
||||||||||||||||||||||||||||||||||||||||||||||||||||
IMEET MESTO RAWENSTWO |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
∂S |
X(x0, t), t |
∂X |
|
|
|
|
|
|
|
|
|
∂X |
|
|
|||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(x0, t) = P (x0, t) |
|
|
(x0, t), |
|
|
|||||||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
∂x |
∂x0 |
∂x0 |
|
|
∂X
KOTOROE \KWIWALENTNO (5.1), POSKOLXKU IZ USLOWIQ (4.10): ∂x0 (x0, t) = 0 DLQ L@BYH x0 R I L@BYH t [0, T ].
lEMMA gAMILXTONA (STACIONARNYJ WARIANT). pUSTX WYPOLNENO USLOWIE TRANS-
WERSALXNOSTI (4.25). tOGDA IMPULXS NA FAZOWOJ TRAEKTORII RAWEN GRADIENTU \JKONALA NA LU^E.
|
|
|
|
Lξ |
|
|
, |
|
|
iNYMI SLOWAMI W TO^KE x = X(ξ, τ), QWLQ@]EJSQ PROEKCIEJ TO^KI |
X(ξ, τ), P (ξ, τ) |
|
|||||||
FAZOWOJ TRAEKTORII |
|
NA KONFIGURACIONNOE PROSTRANSTWO W L@BOJ MOMENT WREMENI τ |
|||||||
TAKOJ, ^TO |τ| < τ0, PRI WYPOLNENII USLOWIQ (4.22) IMEET MESTO RAWENSTWO |
|
||||||||
|
( |
) |
|
|
|
|
|
|
|
(5.3) |
|
|
|
|
P (ξ, τ) = xS X(ξ, τ) |
, |
|
|
|
GDE FUNKCIQ S x |
|
OPREDELENA FORMULAMI (4.18)–(4.24). |
|
|
|
dOKAZATELXSTWO. pROWEDEM DOKAZATELXSTWO DLQ SLU^AQ n = 2 (x R2) (DLQ SLU^AEW
n = 1, I n > 2 DOKAZATELXSTWO ANALOGI^NO). w SILU PUNKTOW 2◦–4◦ ALGORITMA a4 IMEEM
|
|
|
τ |
|
|
|
|
|
|
0 |
|
|
|
||||
(5.4) |
= S˜(ξ, τ) = S0(ξ) + |
|
pH(x, p) |
|
x=X(ξ,τ , dτ , |
|||
S X(ξ, τ) |
|
p, |
|
|
||||
|
|
|
. |
|
|
|
p=P (ξ,τ )) |
|
|
|
|
|
|
|
40