ΚΛΑΣΙΚΗ ΜΗΧΑΝΙΚΗ - physics.ntua.grdris/KLASIKHMHXANIKH061015istosel.pdf · 6 2....
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Transcript of ΚΛΑΣΙΚΗ ΜΗΧΑΝΙΚΗ - physics.ntua.grdris/KLASIKHMHXANIKH061015istosel.pdf · 6 2....
-
1
06-10-15
.
2015
-
2
-
3
Timeo hominem unius libri.
Thomas Acquinas
.
, , .
, .
, . , .
.
-
4
-
5
1.
1.1 1.2
1.3 ( )
1.4
1.5
1.6
1.7 ,
1.8 ,
1.9
1.10
1.11
1.12
1.13
1.14 ,
1.15
1.16
1.17 ,
1.18
-
6
2. , .
2.1 ,
2.2 ,
2.3
2.4 ,
2.5 , , ,
2.6
2.7 ,
2.8
2.9
1) , ,
1( ) / aV r G r
2)
3) ()
4)
-
7
5) Laplace-Runge-Lenz (LRL)
6)
7)
3.
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
-
8
3.14 , Coriolis
3.15
3.16
4.
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
5.
5.1
5.2
5.3
-
9
1.
. , , . . . . () . .
. , , , , , , . .
, , . . () . , , , . . Schroedinger. () ( ) . , . , , . () Schroedinger.
()
= , ddpFt
.
-
10
() .
, . . .
,
ddrp m mt
d
dpFt
2
2
d d, , .
rF mt
mx X my Y mz Z
(1.1)
, ,X Y Z F
.
, ,
2
2
2 2 2
2 2 2
1
, , 1 1 1
, , .
yx zx y z
x y z
mpc
mm mp p p c c c
p X p Y p Z
(1.2)
. () .
, .
-
11
, , .
, , . . , . . , . () ( ). . . . , .. .
, , , . .
, . - . .
-. , , .
-
12
, , , .
, :
1) , . .
.
2) , ( ) . (1.1) (1.2) () , .
3) , ( ). , () () .
, , ( ), () ( ). .
.
, , . , . , ,
, , . : , (local system, ), , .
-
13
. . , , .
Lorentz, , . .
, . .
, ( ) , , , , ( , , )F F r r t
. . , , .
, , , . . , , , .
, . , .
, ,
-
14
. L. A. Pars.
( ) , ( ) , .
, , , (.. ) .
1.1
. . . . .
, . . , () . 3 . , , , ... -. . , , : ) , ) , ) )
-
15
. (), .
( ). . - . , .
( ). . , () 1,2,3 4, 4 41 42 43F F F F
41 42 43, ,F F F
1,2,3 4 4 (. . 1.1). () .
1.1
p m , 2 ,
-
16
dF m madt
dadt
,
, .
2
21
mp
c
.
.
1.2
() m, , F
, . 1.3.
2
2
d d dd d dp rF m mt t t
n ,
2n n n
n 2
d d dd d dp rF m mt t t
(1.3)
n . . (1.6) , , n.
-
17
1.3
, , , .
. .
1.4
. Ox, Oy, . 1.4. t = 0 0 0,x y 0 Ox . t
-
18
x, y. 2 . Ox, 0 = md2x/dt2 Oy, -mg = md2y/dt2. ,
2
2
dd
ym mgt
, (1.4)
, t = 0, 0 0 0, d / d sinyy y y t
2
2
d 0d
xmt
(1.5)
, t = 0, 0 0 0, d / d cosxx x x t . ,
2
1 22gty c t c 3 4x c t c .
, :
1ddy gt ct , 3
ddx ct
t = 0 , 0 2 0 4, y c x c
0 1 0 3sin , cosc c
:
0 0
2
0 0
cos
sin .2
x x tgty y t
(1.6)
-
19
0 , . 90 - . , .
1.3 ( )
( )u () ,
2
2
d d( ) , 1, 2,...,d d
A u a b c nu u
(1.7)
0u u ( ):
0 0d d( ), ( )d d
u uu u
(1.8)
( ) ( )A u A u
( ) ( )u u
,
2
2
d d( )d d
A u a b cu u
(1.9)
( 0u u )
-
20
0 0 0 0dd d( ) ( ), ( ) ( ).
d d du u u u
u u u
(1.10)
. ().
,
, a , a m , b , c k ( ), u , u t ( ) ( )A t F t
.
() .
.
2
2
d d( ) , 1, 2,...,d d
A u a b c nu u
.
(1.11)
() ( ). 1 2 3, ,x x x 1 2 3, ,F F F , 1 1 2 2 3 3 1 1 2 2 3 3, r x e x e x e F F e F e F e
. . (1.11)
. mg . 1.5 -mgy OO'. , OO' 0 Oy. 0
2
2d d( )d d
r rF t m b krt t
2
2
d d( ) , 1,2,3.
d di i
i ix xF t m b kx i
t t
-
21
. , OO'
1.5
Oy . , , . , OO Oy . b k .
1.4
, . . .1.6, 1 2, ,...p p
. j ijF
i. i extiF
, .
1.6
-
22
iF
i . 2 ,
dd
ii
pFt
. (1.12)
exti i ijji j
F F F
extdd
ii ij
ji j
pF Ft
(1.13)
, i :
extd dd d
ii ij i
i i j i ii j
pF F pt t
ext extii
F F
,
tot ii
p p , .
- , , ij jiF F i j
,
0iji j
i j
F
, :
totextddpFt
, (1.14)
ext 0F
:
-
23
tot totd 0 .dp pt
, - ( ) ( ) . () .
1.5
()
c
. 1.7. t A t + t 1 .
1.7
() :
1 1avAA r r r
t t t
.
1AA . 0t , t A :
-
24
0
d ( )( ) lim dtr r ttt t
1A A, 1AA A. e , , . 1.7,
1 1 , 0r AA e AA .
1
0
d d, limd dtAAr re e
t t t
.
e av
r .
c . 1.8
1.8
t A t + t 1A 1 , ( ) :
1ava t t
.
-
25
, c . , 1, ,
ava
. 0t 1A A. 1
A c A. A. , . , . , 0t ava
c. a . :
2
12
0 0
d ( ) d ( )lim lim d dt t
t r tat t t t
.
( ) ( ), ( ) d ( ) / dr r t t r t t ( ) , 2 2d ( ) / da r t t . , , ( ) d ( ) / dA t B t t
n ,
nnd ( )( )
dB tA t
t . (1.15)
! .
()
(1) .
,
-
26
( ) ( ) ( ) ( )x y zr t x t e y t e z t e . (1.16)
, , ,x y ze e e , :
d ( ) d ( ) d ( ) d ( )( )d d d dx y zr t x t y t z tt e e e
t t t t
(1.17)
,
( ) x x y y z zt e e e (1.18)
d dy dz, , d d dx y zxt t t
.
( )a t ,
2 2 2 2
2 2 2 2
d ( ) d ( ) d ( ) d ( )( )d d d dx y zr t x t y t z ta t e e et t t t
, (1.19)
( ) x x y y z za t a e a e a e ,
2 2 2
2 2 2
dd dd d d, , d d d d d d
yx zx y z
x y za a at t t t t t
(1.20)
(2) .
-
27
( )r r t .
( )re t
. 1.9 ( )e t
, re . r ,
( ) r rt e e ,
re e
.
( ) ( ) rr t r t e
1.9
:
dd dd d d
rr
er r e rt t t
(1.21)
( )re t .
-
28
A t B t + t. ( )re t , ( )re t t
. , A ( )re t t
. ( )re t t
( )re t ,
OB OA. 0t , re
1re
, A . ,
d
d dr rr
ee
e
,
0t re 90 ( )re t ,
d dr re e e e
, re ,
. ,
dd d
d d drr ee e e
t t t
,
,
d d dd d drr r e r et t t
(1.22)
d(t)/dt = (t) A O t. ,
d d d drr r rt t
. (1.23)
r ra a e a e ,
,ra a .
d dd dre
et t
,
-
29
22 2
22 2
d d d d 1 d d( )d d d d d dr
r ra t r e r et t t t r t t
. (1.24).
2
22
ddr
ra rt
21 dda rr t . (1.25)
r , :
2 d d, d dr
a r a rt t ,
( re
) .
()
c . 1.10, . s = s(t) c . O s O A, , . t A, s s(t) , t + t s = s(t + t). ,
1.10
-
30
1AA
s , 0 ( 0)s t , ,
1AA
A (. . 1.11).
10
lims
AAes
e . ,
0
d( ) lim dts stt t
d( ) ( )dst e t et
( )t , .
:
2
2
d ( ) d ( )d (t) d ( ) d ( ) d ( )( ) ( )d d d d d d
e t e ts t s t ta t e e tt t t t t t
.
d ( )d
e tt
.
t t + t e ( )e t
( )e t t ( ). A () ( )e t t
. ( )e t
A ( )e t t B
, . 0t ,
d dd d ne e et t
-
31
1.11
ne e
A, A . , ( )e t
( )e t t
, B, 0t A. ne
. ( )e t
, ( )e t
t t t , :
d dd d d d d d ne e et t t t
.
,
2 2
2 2
d d d d d d d( )d d d d d d dn n
s s s s sa t e e e et t t t t s t
dds
, 1 / ( )R t . R A.
-
32
. ne
e
A ne e
Fernet A. ,
22 2
2
d 1 d d( )d d dn n
s sa t e e e et R t t R
(1.26)
( )a t , ( ).
n na a e a e
a , na , . , , .
2 d d d, 0, 2 0d d dA A AA A A A A At t t
A
A
ddAt
.
d d0 cos 0d dA AA At t
, cos 0, / 2 .
-
33
1.6
1.12
B
, B , AA",
. 1.12. . AA". A' . ( ) ( )B B t t B t
0t B
, d dB h e e
B
t, h B
(
). ,
d ( ) dd dB t h e
t t
( )t .
d( )d A
t et , Ae
AA .
sinh B
-
34
d ( ) ( ) ( )dB t t B t
t
. (1.27)
( )t . . B
, ( )r t ,
d ( ) ( )dr t r tt
. (1.28)
B
, ( )t ,
d ( )( ) ( ) ( )d
ta t t tt
. (1.29)
, B A A ,
d 0dt
,
d d 0d dB A At t .
1.7 ,
m ( , , )F F r t .
,
d dpFt
, dd
F mt
, p m .
-
35
dr dt , :
dd dd
F r m rt
dd d = d
drF r m mt
(1.30)
1t 2t
1 1 1 2 2 2 1 1 1 2 2 2( ), ( ), ( ), ( )r r t r r t t t ,
2 2
1 1
d dr
r
F r m
, 2 2 2
1 1 1
2 2 2 22 1
1 1d d d2 2 2
r
r
mF r m m
. (1.31)
212
T m
. dF r
d d ddrF t F tt
( t ). 2
1
dr
r
F r
1r 2r . 1t 2t . , . ,
-
36
2
1
2 22 1
1 1d2 2
ir
i i i i i ii i ir i
F r m m
. (1.32)
( ) ( ) , .
dF t
, ,
2
1
2 2 1 1d ( ) ( )t
t
F t p t p t ,
1 2, t t . .
1.8 ,
, ( )F F r
. 1 2 1 2, r r ,
2
1
dr
r
F r
. (
). , 1r 2r ,
-
37
2
1
2 1d ( , )r
r
F r f r r
(1.33)
1 0r r ,
, ( )V r r , 0r
. , ,
0
( ) ( ) dr
r
V r F r r
(1.34)
ddV F r . (1.35)
, . d dV V r
, ( )V r
(gradient) ( )V r . ( )F V r .
c d 0c
F r
Stokes, nd dS c
A e S A l , 0F
.
F .
2
1
2 22 1 2 1
1 1d ( ) ( )2 2
r
r
F r V r V r m m
2 2 21 1 2 2 tot1 1 1( ) ( ) ( ) 2 2 2
V r m V r m V r m E . (1.36)
.
-
38
. , .
, ( , )V V r t
( , ) F V r t t .
. E T V . .
1.9
. cmR
:
cmi i
i
ii
m rR
m
(1.37)
cm, ir R i
. ii
m M , .
- ,
cmcm
dd d
d
ii i i
i i i
ii
r m pR PtV
t m M M M
(1.38)
-
39
P
.
2
cm cmex2
d 1 d d 1, d d dV P R F
t M t t M
(1.39)
exddPFt
.
. ex 0F
.
(
)
cmdd iiRM p P
t
(1.40)
M .
. - . , , , . . , , .
-
40
1.10
1.15
m p
. O OA r
.1.15 ( ). m O
L r p mr (1.41)
m . L
r p ( , ,r p L
). A F F O N r F
. p O. , ( ),
ddpFt
r ,
ddpr F N rt
d dd dpr F N r mrt t
,
,
-
41
d d dd d d
dL rm mr m mrdt t t t
,
0 ,
d d
dL dLmr Ndt t dt
. (1.42)
2
1
mp
c
,
2
d d d d0d d d d
1
dL r p p pp r m r rdt t t t t
c
. 0N
. .
0N
, L
,
.
1.11
. O, . oiL
i O, ir
-
42
or
O ( ),
o o( )i i iL r r p . (1.43)
i, exiF
ijF
j ,
exi i ijjj i
dp F Fdt
. (1.44)
'
o( )r r ,
o o ex od( ) ( ) ( )d
ii i i i ij
jj i
pr r r r F r r Ft
(1.45) ,
o ood d d d( )d d d d
i i ii i i
L p r rr r p pt t t t
(1.45)
dd
ii i
rp mt
dd
irt
.
,
o oo ex od d d( ) ( )d d d
i ii i i ij i
jj i
L p rr r F r r F pt t t
. (1.46)
-
43
o oii
L L
o o ex( )i ii
N r r F
, ,
o oo od d( )d di iji j
j i
L rN r r F Pt t
(1.47)
ii
P p
.
- ij jiF F
() . o o( ) ( )i j i jr r r r r r
ijF jiF
,
o o o o( ) ( ) ( ) ( ) ( ) 0i ij j ji i ij j ij i j ijr r F r r F r r F r r F r r F .
o ood dd dL rN Pt t
. (1.48)
.
, oddrt
, P
,
. .
o oddL Nt
. (1.49)
. , .
-
44
1.12
..
( )
o i i ii
L m r .
i i . ir
i
A Ai ir r r
Ar ()
A Air A
i . ,
A Ai i ii
L m r o A AL L r P
(1.50)
cmi ii
P m MV , i
iM m
cmV
.
, O P
( !).
i
.
A C .
-
45
cm ci i ii
L m r (1.51)
cm ci iV ,
cmV
, ci i
( ). ci
, . ,
cm c c c c( )i i i i i ii i
L m r m r V . (1.52)
c c c c cc c 0i i i ii i
m r V m r V MR V
cc 0R
() !
cm c ci i ii
L m r .
,
o cm cmL L r P . (1.53)
,
exddLNt
. (1.54)
-
46
.
cmex,cddLNt
. (1.55)
, .
.
2
cm cmex 2
d d dd d dR PF M Mt t t
. (1.56)
, , , . , .
1.13
i
cm ci ir R r (1.57)
cmR
cir i.
cm c cm cd d dd d d
i ii i
r R r Vt t t
. (1.58)
-
47
i i , cV
ci
i . :
2 2cm c
2 2cm c cm c
1 1 ( )2 21 ( 2 ).2
i i i ii i
i i ii
T m m V
m V V
(1.59)
,
cmi i
im r
RM
cm cm c cm c1 1 1( )i i i i i i
i i iV m m V V m
M M M
(1.60)
c 0i ii
m . (1.61)
c ci i ip m ,
, , . kE ,
2 2cm c cm c
2 2cm c
1 1 22 21 1 .2 2
i i i ii i
i ii
T MV m V m
MV m
(1.62)
,
-
48
(, ).
1.14 ,
, C ( . . 1.16), :
cm 1 1 c c1 2 2 c c2( ) ( )L m r r m r r (1.63)
c1 c1 c2 c2, r r , c1 1 c c2 2 c( ), ( )r r r r r r
cm 1 c1 c1 2 c2 c2( ) ( )L m r r m r r .
1.16
1 c1 2 c2 c1 c20, m r m r r r r ,
1 c1 2 c1
2 1c1 c2
1 2 1 2
( ) 0
,
m r m r rm mr r r r
m m m m
(1.64)
-
49
1 2cm
1 2
cm
1 2
( )
( )1 1 1 .
m mL r rm m
L r r
m m
(1.65)
.
,
cm ex ex1 ex2ddL N N Nt
(1.66)
.
2 21 2
1 1 2 22 2
2 2ex1 21 2 1 2 12 21
2 21 2 1 1 2 2
d d, d d
d ( ) dd d
ex
r rm F m Ft t
F Fr r F F F Frt t m m m m m m
(1.67)
12 21F F
,
2
ex1 ex2122
1 2 1 2
d 1 1d
F Fr Ft m m m m
. (1.68)
ex1 ex21 2
0F Fm m
, (1.69)
-
50
2
122
dd
r Ft
. (1.70)
.
, ( ) .
1.15
1.17
A
O, . 1.17, P
, P r A
r B . C A 1 2r r r
,
1 2 1 2( )P r r A r A r A , (1.71)
2 0r A 2r
A
. ,
1P r A . A
,
, A
.
-
51
1.16
. .1.18
np On, n n ( )p e r A .
On C ( A
). O1C
On. A
1 n, , A A A
. 1A
O1C, nA
On A
On . A
O1C. ,
1.18
n n n 1 1 1 n( ) (OO O ) ( )p e r A e C A A A (1.72)
n n 1 1 1 1 n 1 1 1 1 nOO OO OO O O Op e A A A C A C A C A
1 1 1 nOO 0, OO 0A A
-
52
n n 1 1 n 1 n 1 n n 1(OO ) (OO ) (O C ) (O C )p e A e A e A e A .
( ) ( )a b c a b c (1.73)
2 ,
1n 1 n 1 1(OO ) ( OO ) 0e A e A
ne 1OO
.
1n (OO ) 0e A .
,
n 1 n n n 1 n n 1(O C ) ( O C) ( ) O C 0e A e A e A .
,
n n 1 1(O C ) (O C)p e A A . (1.74)
O . A
nA
On.
-
53
1.19
. 1.19, A
O
A Ox, Oy, Oz. ,
o
( ) ( ) ( )
x y z
x z y y x z z y x
x y z
e e ep r A x y z e yA zA e zA xA e zA xA
A A A
(1.75)
op x, oxe p
:
, , x z y y x z z y xp yA zA p zA xA p xA yA (1.76)
A
() , .
1.17 ,
( ) ( ) rF r F r e , .
-
54
N
,
( ( ) ) ( ) 0r rN r F r F r e F r r e (1.77)
r re ().
,
od 0dLNt
(1.78)
o L mr . r ,
o( ) ( )mr r r L . (1.79)
,
o( ) ( ) 0mr r m r r r L . (1.80)
r oL
.
oL
.
Oz ( ) oL
,
,
o o oz 0zr L L r e L , (1.81)
zr e r Oz z . 0z
. Oz . , z () oL
.
-
55
1.18
, . 1.20. , , .
1.20
, 2 1 1F
1 2 2F
.
, 1 2F F
. . . , . 1.20, 1 2,N N
-
56
1 2,T T , , , () (dry friction).
( )
1 2N N N , N . 1 2T T T , T .
.
, .
.
, ) , , ) , .
, :
) .
kT n N (1.82).
( ) kn . , .
) .
) .
max sT T n N (1.83)
-
57
sn , .
kn . s kn n .
s kn n n . s kn n , , . .
2. . . 2.1 ,
, . r . 1 2 . (2.1)
1 212 122m mF G e
r
(2.1)
. 2.1 .
-
58
2.1
G , SI 11 2 26,672 59 10 Nm / kgG . 1 2,m m
, . r . 12e
12r
, 1 2 .
12 121212
r rer r
12 0r r
12r
1 2. .
21F
2 1 1 2 , . (2.2)
2 121 212m mF G e
r
(2.2)
. 12 21=-e e
12 21F F
(. 2.1).
- .
-
59
1 212 2 13
2 1
( )m mF G r rr r
(2.3)
1 2,r r
, . 2.1.
2.2
1,2 2 1.
2 121 1 2 1231 2
( )m mF G r r Fr r
.
Cavendish ( mm, 50 m). 1510 m , , , .
( ) . . ,
-
60
/ /1 2 1 2 1 2( ) (1 e ) er rm m m m m mV r G G Gr r r
.
() Yukawa.
, 3 27,2 10 , 2,00 10 m .
, . , , , . Eodvos.
. ( / ) .
( ) .
. .
1
n
ii
F F
.
.
. . .
. 2.2 ( ) . 1 , dm . 2m
-
61
2 12 3
2 1
( )d dr rF Gm mr r
dm 2m .
2 12 3
X 2 1
( ) dr rF Gm mr r
. 2 1R r r
2 23 2X X
d dReRF Gm m Gm mR R
Re
1 2 R
R
R
, ( RR Re ).
, V . d dm d .
d dm S dS . , S . , , d dm l , dl . , c .
.
. , ( , am ) ( , pm ). 1 2 ,
a1 p212 2
m mF G
r
. 2 1 ,
-
62
a2 p121 2
m mF G
r
. - 21 12F F
a1 a2a1 p2 a2 p1
p1 p2
, m mm m m mm m
.
, 1 ( ). m .
m ( ) , ( c ) F ma . 1 2 a .
1 212 22
m mF G m ar
. 2 2m
2m .
1 22
2
m ma Gr m
.
2 . 22
mm
. , 1 , m , ,.
1
, , M r . .
. 2.3. y M x .
-
63
2.3
. d dm y M dF
.
2 2
d dd m xF GM GM
.
, xF F . ( ) .
2
dd d cos cosxyF F GM
, 2tan , d dcosy ryr
, / cosr .
d cos dxGMF
r . ,
, 0 /2
-
64
/2 1
0 0
2 cos d 2 d(sin )
2 .
xGM GMF F
r rGMF
r
2.2 ,
, . , , . , . . . g . ( , )g g r t . g g . ,
Fgm
, F
m
. m . , m , m . 0m ( ) . . , .
( , ) m . tm . r
m
tm . m
tm , t2t t
rmmFg G e
m r m
, . 2.4.
-
65
2.4
g . g , , . . . . , , , .
, , . . . ( ) . 2.4. .
-
66
. .
gF
g m gF mg , ga
gF mga gm m
.
g g .
g
dS
d ( d ) d cosg S g S = , . 2.5.
2.5
Gauss . , ,
4Gm , m . . , . 2.6.
-
67
2.6
m S g .
dS
, d ( d ) d cos(- ) d cosg S g S g S =
, 0g , ( ) g . d cos dS S , dS r . d dg S .
2mg Gr
2dd dSGm Gmr
, d
d ,dS S . d dGm . (). dGm =- . 4 , 4Gm .
. 2.7.
1S 2S . c . c
-
68
S . 1S
2.7
1 0, 2S
2 0 . . 2.7
, 2 1 0 . ,
1 2 1 1+ +(- ) 0 = dGm =- =0. 4Gm . g .
4Gm .
(divergence) ( )r
( ) 4 ( )g r G r .
-
69
, .. . ( ) 0.g r
( , , ) .yx zgg gg x y z
x y z
( )1 1( , , ) .r zgrg gg r z
r r r z
2
2
(sin )( )1 1 1( , , ) .sin sin
r zgr g gg rr r r r
. , . 2.4 . , . . . . , , .
2.3
, . ,
-
70
. m , . 2.8.
2.8
tm Ar
Br
B
A
B B
ABA A
( d ) d cos ( )dr
r
W F r F r F r r .
B
A
AB t t2B A
d 1 1r
r
rW Gmm Gmmr r r
.
. , . , , .
(. 2.2),
tt3 2
X Xt
( )d dRr r eF G m Gm mRr r
.
-
71
t3 2
t X Xt
( )d dRr r eFg G m G mm Rr r
.
Ar
Br
B B B
AB t tA A A
( d ) ( d ) ( d )W F r m g r m g l .
l d dll e l , d d dlr l e l
.
, .
.
0)d()d( cc
lgmlF , m .
( )U r m r Ar
,
A( ) ( , )U r W r r
. m r , A( ) ( , )U r W r r
.
, , .
, . . . . . . .
-
72
, , . , .
r ( ) ( )( ) U r W rV rm m
m
r ( )U r m , ( )W r .
B B B
A A A
( d ) ( d ) ( d )U W F r m g r m g l .
( )g r m ,
B B B
A A A
1( ) ( d ) ( d ) ( d )V r F r g r g lm
.
, . ( )A r
c
lA 0)d(
,
.. c .
c
lg 0)d( .
(rotation) ( )A r
A
, , 0A
.
. 0A
, . .
, 0g .
-
73
( , , )
x y z
y yx xz zx y z
x y z
e e eA AA AA AA x y z e e e
x y z y z z x x yA A A
.
( )1 1( , , ) z r z rr zA rAA A A AA r z e e e
r z z r r r
.
(sin ) ( ) ( )1 1 1 1( , , )sin sin
r rr
A rAA rAA AA r e e er r r r r r
m 1r
2r
12 1 2 1 2( ) ( ) ( ) ( )W U r U r V r V r m
.
, ( 9.9) r
X X
1 1( ) d dV r G m G mr r R
.
.
() .
, . .
.
1 1
n n
i i ii i
U U V m
.
X
dU V m .
-
74
, .
.
. . , 1 0W , .
1 2212
m mW Gr
.
1 2,m m ,
. 1 23 313 23
m mW Gmr r
3m
1 2 m m . n ,
1 1
1 1 1 1 1
n n i n ij i j
i ii i j i jij ij
m m mU W G m G
r r
.
,i j , 1 n
1 1
12
n ni j
i j iji j
m mU G
r
.
i j , i j .
1
nj
ij jij i
mV G
r
i
.
1
12
n
i ii
U G mV
.
-
75
,
X
1 d2
U G V m .
, .
2
12 2 11
( d )W F l K K
.
1 2,K K 1 2 , . 12 1 2 1 2( )W U U m V V , m .
1 2 2 1U U K K 1 1 2 2K U K U . E K U . , 1 2E E . .
, .
, , , . , , ( ) .
, , . , , .
dW F
dr d dW F r .
, d d dF mg W m V mg r
1 2 2 1d d , ( )V g r V V V V V .
-
76
d d d cosV g r g r ,
,dg r . ()
( )V V r , d d d cosV V r V r
,dV r . g V
. g , g , . 0 cos 1 .
( , , ) yx zx y zAA AA x y z e e e
x y z
.
1( , , ) r zA A AA r z e e er r z
.
1 1( , , )sinr
A A AA r e e er r r
.
m
( ) ( ) mV r V r Gr
. 2
dd r rV mg V e G er r
.
.
1
g .
-
77
2.9
m R (. 2.9).
33
4mR
.
, r R . 1S r . 2.19, g g . 1S
1 1
( d ) d 4 4S S
g S g S Gm Gm = , m m .
1
2( ) d 4 ( )4 4S
g r S Gm g r r Gm
2 2( ) ( ) rm mg r G g r G er r
. g
r re
d d rS Se ( d 0)S .
-
78
. ( )r .
, ( )r R 2S
, 2 2
( d ) d 4S S
g S g S Gm = .
r 3
3 3
4 3
mrm rR
. 3
23( )4 4
mrg r r GR
3( ) rmg r G reR
.
.
1 2,m m
1 1 1 2 2 2( ), ( )r r 1 2,R R . 1 2,r r .
1 2r R R . 1 22m mF G
r
. 1 2,m m .
.
. 2.10. im
2.10
-
79
im
1m ( 1O ) .
im . im
1m 1O . ( )im
1m . , , 2m 2O .
. 1 212 21 122m mF F G e
r
1 2O Or .
2
m . . 2.11.
.
2.11
-
80
. 1r . r . sinR . R 2( sin ) dR R .
2
2
2 sin d4
Rdm mR
1d sin d2
m m .
1 1
d 1 sin dd2
mV G Gmr r
.
2 2 21 2 cosr r R rR . ,r R ,
1 12 d 2 sin dr r rR , 11
dsin d rr rR
. 1d d2GmV rrR
.
1r r R r R ,
1( ) d 2
r R
r R
Gm mV r r G r RrR r
.
.
, 1r R r R r
1d 2R r
R r
r r
, ( ) mV r G r RR
.
, .
( )r r , r R , . . . . ()
( ) mV r G r Rr
.
( )r . ,
. . 2.12.
-
81
2.12
r . , 1dr , 1r dm ,
21 11 1
d d 4G GV m r drr r
.
, 1 1 2R r R .
2
1
2 21 1 2 1( ) 4 2 ( )
R
R
V r G r dr G R R .
1 2, ,m R R .
d ( )( )d rV rg V r e
r r
0g .
2.4 ,
() . ,
-
82
. , . 2.13. . .
2.13
. 2.14 .
2.14
-
83
, ( )g r , (gradient) . g V
. V
. (.. ) . (.. ). . 2.15 , .
2.15
-
84
, . , . .
3 rkA er
.
.
1
. 2.16. ( )V r
g V .
. .
-
85
2.16
. ( )V V r . , . 2.16.
( ) ( d ).r
V r g r
.
( r R ) ,
2 0mg Gr
.
2
d( ) ( d ) ( d ) .r rr r r
rV r g r ge re Gmr
( ) mV r Gr
, r R .
r R , ( ) ( d ).r
V r g r
, ,
1 2( ) ( d ) ( d ) ( d ), .R
r r R
V r g r V V g r g r r R
( 2 )V
r R . 2mV GR
.
( r R ) 2
mg Gr
m r .
34 3
m r
32 2
1 2
4 d 4 2 d ( )3 3 3
R R
r r
r rV G G r r G R rr
.
, 34
3
m
R
-
86
2
1 2 32Gm rV V V
R R
.
2.5 , , ,
, . . re
, , rF Fe
, F , ( ).
. 2.17. m rF F e
, . ( )
dd
F ma m mt .
2.17
-
87
c , . .r F mr
, .
L mr .
d d 0d dL rm mr m mr r Ft t
.
0r F , ,r F
0 , ( sinrF ). d 0, .dL Lt
, . ( ). , L
,r , r ,r L
,
.
, . .
d dd drr e r et t
L
2d d d0 d d dr r rrL mr mre e r e mr e et t t
.
re e
e
. .
2 d d
L rt
.
-
88
2.6
. ( )V V r
( )A r ( )
( ) ( )A r V r .
1 1 , , ,sin
, , .
r r r rV V VA e A e A e e e e A A A Vr r r
r
0A A 0V V
r .
, d ( ) d ( )( ) ( ) , ( )d dr rV r V rV V r F e F r e F r
r r
.
( ).E T V r
() . , . 9.18,
() . 2 12 2 1 1 2 1d d, , d d
F m F m F Ft t
.
2 1r r r
, 2 22
2 1 2 1 2 222 2 2
2 1 2 1 2 1
d dd 1 1d d d
r r F F F Fr Ft t t m m m m m m
.
2 1
1 1 1m m
, ,
2
2 2
dd
rFt
.
()
2 1r r r
. r 1 2F
r
1 2,m m .
-
89
. , , .
2.18.
2.7 ,
M . 2.19. m M , . ,M m . M m M .
-
90
m 22
22
d d 1 d dd d d dr
ra r e r et t r t t
.
2.19
m ,
2 , 0rkF e k GMmr
.
222
2 2
d d 1 d dd d d dr r
k re ma m r e r er t t r t t
,
2 d d
L rt
22
2 2
d dd d
k r rmr t t
. 2 2
2 2 3 2
d 0d
r L kt m r mr .
( ), ( )r r t t , ( )r r .
2 2
2 2 2 5
d d d d 2 d, d d d d dr r d L L rt t dt mr t m r
-
91
2d dd dr r Lt mr .
22 2 2 2
2 2 2 4 2 5
d d d 2d d d
r r L r Lt m r m r
.
22 2 2
2 4 2 2 3 2
d 2 d 0d d
L r r L km r r m r mr
. 1( )( )
ur
2
2 2
d 0.d
u kmuL
,
0 2cos( )kmu AL
, A = .
0 . .
2 2
21 d d 2 d d
r kE m rt t r
. u
2221
2L duE u kum d
. 0sin( )du Ad
2
2 2
21km ELAL k m
. 2
02 2
1 21 1 cos( )km ELur L k m
.
( )r r (, , , ).
2
2
21 ELk m
.
2
01 1 1 cos( ) , Lsr s km
.
E ,
-
92
0 1 , ( 0, )0 1 , 0 1 ,
EEE
E T V T V T V .
( )r r .
. A + 0 0r r
r 0 + (. 2.20, ). (. 2.20, ). +, .
2.20
-
93
0 , +,
1 0 1sr r
0 2 1sr
,
2 1r r . 2 1r r .
, 0 ( 1) 1r , , . () . 0
2r
. 1 r r .
2 r r .
0
11 cos( )11
.
r r
r r
r rr r
. 2.21 .
2.21
-
94
2
1Lkmr
.
( 0 , r r ) () .
Lmr
, 2
1mrk .
2
2c
c c
mGMmr r
.
0, cr r , 2
c
GMr
.
2
1.c
20
0 20 0
/.
1 / 1 cos( )c
c
r r
0r 0 , 0 ),( 00 rr
0r . , . 0r ( ) .
2
2
/.
2 /c
c
r r
. 2.22 0, rc , ()
0 .
. c 20
, c 20 ( )
. , c 20
-
95
, . )0( 00 rr (), )0( 00 rr 0 . c 0 cr . , c 0 , , )0( 00 rr () .
2.22
. 2.23 m . (
) d1 d 1d d sin sin
2 d 2 drAA r r r
t t
.
-
96
ddrt
d 1 ( sin )d 2A rmt m
.
.
, d d 2A Lt m .
2.23
, ( ) . , 12t 12A
, 12 122LA tm
12 122mt AL
.
-
97
ab ,a b .
, T . 2 mT abL
.
, 21ba
,
222 1 .maT
L 2a r r
2
2
22(1 )
Lakm
.
22 34 .T a
GM
, , . .
1
, .
2
, .
3
.
( ) , . , , . . .
() . ( ) . - .
-
98
. . , , . .
. )(rVV
2 2
2 3
d d ( ) .d d
r L V rmt mr r
2
2( ) 2LV V rmr
.
)(rV 2
22Lmr
. 2
2
d d ( ) .d d
r V rmt r
()
2
3
Lmr
.
2 2 2
21 d 1 d ( ) 2 d 2 d 2
r r LE m V m V rt t mr
.
0)( rkrV ( )
. 2.24 .
-
99
2.24
V 22
m 21
LmkV .
0E , 0/21 22 ELmk
r . 22 /21 LmkE ,
0dd t
r
kmLrc
2 .
-
100
1
( ) m R . . . M , RR . . m 1037,6 6 R ,
24 1098,5 M kg.
2211 kg/Nm 10670,6 G .
.
R RGMmmE 21 2
1
, 02 E .
021 2
21 RGMmmEE .
RGM 2 .
2R
GMg , 2gR .
6
2411
1037,61098,510670,62
m/s1,1 m/s 104 =11 km/s.
R
GM1
RMG2
2 12 2 .
-
101
2
( ) ) . ) ; . . kg 1099,1 30H M
- m 1049,1 11 R . 365 .
)
skm 42
sm102,4
sm
105,110210722 4
11
3011
H
HH
RGM
.
) ( ). . . ,
skm 7,29s
m360024365101,4922 11
TR .
. ,
s
km)7,297,43( .
, , , .
3 , 3 2 .
-
102
3
, m , ( M ) , . , ( , ). ( ) . .
22
rMmGr
m , rMmGmUKE 22
1 .
rMG 2 r
MmGK 21 . UK 2
1 ,
rMmGU , . r
MmGE 21 .
.
, r . r . , () .
.
2.8
. , . , . , .
-
103
. . (International Standards Organization, ISO) (SI), . , . . , . g , Coriolis .
ggg , , gg
(
) () .
g , , . .
2.9
1.
0,)( 1
Gr
mGrV
M , 24dd
RM
Sm , . 2.25.
-
104
2.25
)(rV r . dR , x ,
11
d2d DRGmGdV . cos2222 RrrR , sinRD ,
dsin2d2 Rr . dd rD .
d2d rRGV .
Rr 0 rR 1
rR 2 .
rR
rR
RGrV dr
2)( ,
RrrRrRrR
MGrV
0 ,1
)()(12
)(11
.
Rr Rr 1
Rr 2
Rr
Rr
RGrV dr
2)( .
-
105
RrRrrRrR
MGrV
,1
)()(12
)(11
.
. 0 .
0 , Coulomb. 0 . .
Coulomb ( Gauss) . Cavendish . , .
2.
(Big Bang), . , . M , . 2.26. ( ) , )(tR
-
106
2.26
. () m .
m , )(2 tR
GMmF .
m )(d
d2
2
tRGM
tR
. tR
dd ,
mEtRGM
tR
)(dd
21 2
mERGM
tR
2
dd
21 mE .
mE . . ,
).()(34 3 ttRM
crtatR )()( , cr ,
-
107
(comoving) . )(ta . ,
tata
tRtR
dd)(
dd)( ,
mcc EtatGrar )()(34
21 2222 .
)(12)(
38
22
2
tarE
tGaa
c
m
.
Friedmann ( ) . . 0a ( ) . 0a . ( ). () mE .
0mE , , 2a . 0mE ,
)(ta
0max cmrE
GMa .
0)(dd
22
2
RGMrta
tR
c , ,0a .
0mE . 0a t 0 .
, ( ), . ( ) . ( ) .
3. ()
, , )(,)( rfFFerfF rr
. ,
22
dd ,d
dmr
Lt
tmrL . ()
-
108
, mrf
rmL
tr )(
dd
32
22 . cr
crr 0dd2
tr , m
rfrm
L )(32
2 .
, () , () . xrr c , crx .
0)()(d
d3
22
m
xrfxrm
Ltx c
c
. Taylor
x ,
cc
c
c
c rxr
rx
rxr
311
)(1 3
3
3
3
xrfrfxrf ccc )()()( , 0d
d)(
x
c xfrf .
0)]()(3[ xrfrfrxm ccc .
0 Dxxm . 0D x , . 0D , .
0)(3)( c
cc rf
rrfD . .
0 ,)( kkrrf , .3
2 .
1 , 3 4 .
4.
3.
: )(dddd22
rftr
trm
, 0dddd 2 tmrt , rerfF
)( .
, , . ( ), ,
-
109
crr , 0dd
2
2
tr . 2)( r
GMmrf .
, dd2 Ltmr .
2322
cc rGM
rmL .
0dd
232
22
rGM
rmL
tr .
xrr c , crx . x
01)1(
dd
22
33
22
ccc
crxr
GM
rxrm
Ltx . Taylor
crx .
c
c
rx
rx
311
13
, c
c
rx
rx
211
12
.
02131dd
23
22
cccc rx
rGM
rx
rmL
tx . crGMmL 22
02131dd
22
2
cccc rx
rGM
rx
rGM
tx , 0
dd
3
2 x
rGM
tx
c
.
3c
r rGM .
2dd
cmrL
t crGMmL 22 3d
dc
rGMt
,
r . ( ) , , , .
.
5. Laplace-Runge-Lenz (LRL)
0 ,)( ,)( 2 krkrV
rkrf , (LRL)
( ), .
-
110
, rrmkLpA . r L
p .
.
LRL Lenz . , Pauli Schroedinger, Heisenberg ( Born). Schroedinger .
,
rrLpmk
)(1 .
mkA .
trmp
dd ,
tp
rrk
tLprL
dd ,0
dd , 3
.
cbabcacba )()()( .
A
rtr
rmk
tr
rmkprr
rkrt
rrmk
tr
rmk
tLpLt
ptA
dd
dd0)(
dd
dd
dd
dd
dd
232 .
rtr
rmk
tr
rmk
trrrt
rrrkmrt
rrmk
tr
rmkprrpr
rk
tA
dd
dd
dd
dd
dd
dd)()(
dd
22
323
.
rtr
rmkrt
rrrkm
tA
dd
dd
dd
23
. t
rrtr
tr
trr
dd2
d)(d
d)(d
dd2
22 .
0dd
dd
dd
23
rt
rrmk
trr
rkm
tA
.
A
, kmA .
LRL . 2.27.
-
111
2.27
rrrrLpmkrr )(1cos .
)()()( bacacbcba
2)()()( LLLprLLprrLp .
rmkLr
2cos mk
Lr2
)cos1( .
)cos1(1 2 Lmk
r . .
. mkA . 0 , A
,
. 2.26.
. 21 22
Emk
L , E
. EmLkmEmk
LmkA 22222
221 .
-
112
6.
. , , , . . .
, a ( ), a . a . () am . , .
, , , , . . .
. N , , )( ji rrF
.
g .
jiNj
jiii
i rrFgmtrm
,...,2,1
2
2
)(dd , Ni ,...,2,1 .
, ir , ( ga ), ir
. 2
21 tgrr ii
. jiji rrrr
,
jiNj
ji
jiNj
jiiii
i rrFrrFgmgmtrm
,...,2,1
,...,2,1
2
2
)()(dd .
gmi
. ( )
-
113
, . , .
. . ( ) , , . , . , , . . . , .
( ) . , () .
.
, , , () . . , .
, / , .
.
-
114
)
. . . 2.28 , , 1 2 . ( ) g . 1,2 z 0g . . .
2.28
. zz O,O , 1f . g z . 1, 2 g . 1,2 12z , . 2. ( 1,2) , ( 2 ) .
-
115
,
. czt 12 ,
t . 2 gt
cgz 12
( ). () Doppler , c ,
212
1
21
cgz
c
fff
2
1212 1 c
gzff . g ,
, 12 ff ( ), g 12 ff , .
2
1212 1 c
VVff , 21,VV
1,2 .
z Doppler , .
( ) .
. hfE , ,
chf
cEp . fch ,, Planck,
.
g gchfF 2 . c
. tpF d
d
tf
ch
chfg
dd
2 . td
tcz dd , zcg
ff
zf
cfg dd ,
dd
22 ,
1zz , 1ff , )(1ln )(ln 121
112
1zz
cg
fffzz
cg
ff
. 2zz
2ff .
12 ff )1ln(1ln1
12 xxfff
, 1
1
12
fffx .
-
116
,
212
1
12 )(c
zzgf
ff
.
. , g () 1m 1 ( . 2.28 , z ). 1f . 2m 2. . (
) 1 211 chfm 2 222 c
hfm .
,
222
22
22
2121
12
21
1222
2112
1 gzchfmc
chfmgz
chfmc
chfmgzmcmgzmcm
.
2 12
22
21
1
2 )(11
1
czzg
cgzcgz
ff
c
VVff 1212 1 .
)
, . () . . , . ,
-
117
. .
. . 2.29.
2.29
( zx O ) . 0t ( ) , ),( 00 zx . , , zx O c , . xzO . .
. .
-
118
() . . 2.30. ( )
2.30
. 0ry =. 0r . ( ) ,
g 22 chfg
cEgF
.
yF ( ) ( ), 0rx ,
22/32200
22/1220
022
02/122
0
02 c
hfxr
rGMchf
xrr
xrGM
xrr
chfgFy
.
M .
)0( yp y ,
xFccxFtFp yx
yyy d1dd0 ,
2/322
02
0 dxrx
chf
cGMrpy
-
119
pcrGM
chf
crGM
xrx
chf
cGMrpy
02
002/322
02
0 22d2
, 2chfp .
20
2tancr
GMpp
y .
87,0 0,87/3600 . 1,75 , .
.
, .
, . .
7)
( 23 %) . . .
, , () , ,
r , , 0 r R
0, R r .
, , 0 r . 20 .
,
-
120
. 2.31. .
2.31
.
, .
) ( m ) , .
Frm
2, Rr 23 /3
4 rmrGF ,
22 ,34 RrrG .
,Rr 23
/34 rmRGF
2 3 4 1 , 3
G R r Rr
.
-
121
r , , . 2.32 ( ). . 2.31 .
2.32.
) .
rrG 0 ,34 2
Y2 .
2 2 2Y 4 4 , 3 3
G r G r r R
2 2 3 2Y 4 4 / , 3 3
G r G R m r r R .
( . 2.32) , r .
-
122
1. . , . 4 . 3 .
: a .
3.
, , . . . , , , . .
3.1
.
-
123
3.1
. ( ) . . 3.1
B Ar r AB (3.1)
B Ad d dd d dr r ABt t t
. (3.2)
AB
,
d 0dABt
,
B AB Ad dd dr rt t
.
,
-
124
2 2
A A B BA B A B2 2
d d d d, , d d d d
r ra a a at t t t
. (3.3)
, a . , .
3.2
.
3.2
, . , , . . . . 3.2 .
-
125
ddt .
. XDC AB. x . H
2
2d d d, d d dt t t . (3.4)
AB AB.
3.3
B . r C . 3.2 r .
ddA At
(3.5)
,
ddr rt
. (3.6)
(DC) , (DC) e .
2
2
d d d d d ( )d d d d d z
r ra r e r rt t t t t . (3.7)
-
126
2d ( ) ( )d z r z z r
a r e e re e et (3.8)
2 2 nd dsin sin (DC) (DC)d dz
a r e r e e e et t .(3.9)
ze e sin
ACB . 1.20, sinz re e e
. , ze e
n ze e e , . 3.2.
,
2d
d (DC) na e e
t
, (3.10)
.
3.4
. . . .
3.3
-
127
, A . 3.3 , , . .
3.4
. I II . 3.4. , ( ) III. 1B 2B II. 2B
B1A1 2 1B A 2 2B A , II. . 2.4
1 , , . , .. 1 1B C , . B1 A1 2A A1B1 2 3A B 2 1 II.
-
128
. . -, .
, . () . . .
, . . . 2.5. MI M I AA' BB . AIB, A'IB' , .
3.5
I AIA BIB' , = AIA' = BIB'. AIB = AIA' AB A'B' AIB A'IB' .
-
129
, ( ).
. . , . ( ) .
3.5
3.6
B Br r A , r AB
. ,
AA Bdd d,
d d drr rr r r
t t t
. (3.11)
-
130
A BAV A
A , BA
A,
BA AB . O, A, B
. z . 2.6
, ddt . , ()
() () .
.
3.6
. . . 3.7 A B A
B
. AB . .
3.7
-
131
o ,
OA A , o
OA. o OB
B . o 0
. O , . A B(OA), (OB) . O .
, .
3.7
. 3.6 ,
2 22 2
B AA BA2 2 2 2
d dd d , d d d d
r rr r a a at t t t
(3.12)
Aa BAa
. ,
BA c ca a e a e e
( ) AB r
ce AB r
. ,
2
2BA BA c
d d , d d
a r a rt t r
. (3.13)
.
-
132
ddt
. .
3.8
I, II, . 3.8. . . 1A .
1B .
3.8
-
133
1 1O,A,B,A ,B , 1 1 1 1OA=OA =OB , AB=A B . O 1A,B,B ,
C. 1OCA,OCB ,OCB OC 1 1AC=A C=B C. 1CA=CB=CB . 1 1 1ACB,A CB,A CB
. 1 1ACB,A CB . ACB= OC A, B 1 1A ,B . , O I II OC . I II, , . . . = d/dt. , ( )t
. ddt
, .
3.9
, . 3.9, () OA. KC OA,
-
134
3.9
. C O . , d
drrt
KOC
OA, C. , (KC) . ,
d d dd d d
ra r rt t t , (3.14)
, ddt .
a C ( ) r r . KC, .
3.10
. .
-
135
3.10
. 3.10 () I III. , , I II B B1 III. II III. B1 , . B1. B1 . ( ) , . . , . . . .
. , , . C,
C P CP . (3.15)
-
136
C, C ,
( P) P (
) CP
. CP CPr
CPr
C. P
. C ,
CPPC P CP P CP CPddd d ( )
d d d da a a a r r
t t t t .(3.16)
3.11
, Ox'y'z', O'x'y'z' Oxyz Oxyz. Ox'y'z' , Oxyz , O'x'y'z' Oxyz Oxyz . O'x'y'z' ( ) Oxyz. .
3.12
G
, ,
-
137
, . , . , . adG
( ) rdG
, . , G
. . G
tdG
,
(transport) . G
rdG
. "" . ,
a r td =d +dG G G
(3.17)
, G
( ),
. G
,
a r t
d d dd d dG G Gt t t
. (3.18)
( ) . G
.
. , . , G
,
t
ddG Gt
. (3.19)
-
138
a r
d dd dG G Gt t
. (3.20)
.
3.13
. 3.11. M t r Oxyz. AB (O'x'y'z' t). t + dt 1M .
3.11
-
139
1 1A B . Oxyz
M 1M . AB
1 1A B .
, G
r
a r t
d d d , OO O Md d dr r r rt t t
. (3.21)
r
ddrt
, OO
,
rr r
d dO Md drt t
, (3.22)
r
O x y z . . 3.11 .
a r t . (3.23)
( ) . , t
. , O' , t O M r
t O a r O, r r . (3.24)
, ( ) ,
-
140
, ( ).
3.14 , Coriolis
, a r t .
. ,
a a r a t a(d ) (d ) (d ) (3.25)
. 3.12 ,
r a r r r t(d ) (d ) (d ) t a t r t t(d ) (d ) (d )
(3.26)
a t tr rr ta t r
d d dd dd d d d dt t t t t
. (3.27)
, , ( ) , . , :
tra r tt r
ddd d
a a at t
, (3.28)
, Coriolis ca
( Coriolis),
-
141
a r t ca a a a . (3.29)
,
trct r
ddd d
at t
. (3.30)
Coriolis .
rt
ddt . 3.12. Oxyz
.
M t t + dt . AB 1 1A B . r
M M t. t + dt r
1 1A B M' . . r t(d )
3.12
-
142
M' r . 3.12.
r r
t. ,
r rddt (3.31)
.
. tr
ddt . 3.13.
t
O'x'y'z'
3.13
-
143
M , t.
, O' ,
t O r . (3.32)
M ( ) dt M' M' t ,
t O r . (3.33)
,
r r(d ) ( ) MM dr r t . (3.34)
t rr
ddt . (3.35)
rt
ddt ,
trc rt r
dd 2( )d d
at t
. (3.36)
,
-
144
a r t ca a a a (3.37)
a r t r2( )a a a .(3.38)
r 0
, Coriolis . Coriolis r
. , 0 . c 0a
, , a r ta a a
. ta
, . (. 3.13),
t Od + ( )d
a a r rt (3.39)
a r O rd + ( ) 2( )d
a a a r rt
(3.40)
a r O( + )r (3.41)
( )r . dd
rt
.
3.15
. . m
-
145
,
2
2
dd
rF m mat
.
. , aa
( ) ra
( ). ,
a r t ca a a a r a t ca a a a
. (3.42)
2
,
r t cF ma ma ma t c rF ma ma ma
.(3.43)
t cF F ma ma :
rF ma .(3.44)
, . F
F
tma
cma (
) . , F
, . , . (, ) , 1 2 1 2( , )F F r r
. ,
-
146
t Od ( )d
a a r rt , (3.45)
c r2a . (3.46)
F
,
t c
O rd ( ) 2( )d
F F ma ma
F ma m r m r m at
. (3.47)
, , ,
( )r ) ( ) .
3.16
( ) 0, d / d 0t , Oma
. ,
O rF ma F ma .(3.48)
, Oma
. Oa
( ), .
, O 0a
, r aF F ma ma . ,
, ,
( ) . . a ra a
(
-
147
). .
Foucault, .
4.
. .
4.1
, . ( ), 2i im r , im ir .
. On , 2n i i
iI m r . , 2n d
V
I r V
. . ,
2n dS
I r S , 2n dL
I r l (4.1)
K, In M , In = M K2. K .
-
148
4.2 . 4.1. Cxyz C . n Cz x = l Cx. im Ai . A Bi i Cz. B Ci i Cz n . C Ai i Ai n. B A Ci i i
4.1
xCy. im n
2(A C )i i im . ,
-
149
2n (A C )i i ii
I m . (4.2)
B A Ci i i
2 2 2(A C ) (A B ) 2(A B ) cosi i i i i i il l ,
2 2n (A B ) 2 (A B )cosi i i i i i ii i i
I m l m l .(4.3)
(A B )cosi i i ix ( ix Ai ). C z ,
(A B )cos 0i i i ii i
x ii
m M ( ) 2
c (A B )i i ii
I m C z, 2n cI I Ml . , n Ml2, M l ( Steiner ). .
4.3
1.13 :
2 2cm c1 1 .2 2 i ii
T MV m
M ci i , cmV . 3.10 , ,
-
150
c ci ir (4.4)
. . ci iR iR i ( ). ,
2 2 2cm1 12 2 i ii
T MV m R . (4.5)
, cI ,
2c i ii
I m R (4.6)
,
2 2cm c1 12 2
T MV I .(4.7)
. . , , ( ) . .
-
151
4.4
, O, ,
i i ii
L m r .(4.8)
,
i ir , O. ,
( )i i ii
L m r r .(4.9)
Ox, Oy, Oz ir . ,
.
i i x i y i z
x x y y z z
r x e y e z ee e e
(4.10)
,
x y z
i x y z
e e er
x y z
(4.11)
( ) ( ) ( )i i x y i z i y z i x i z x i y ir e z y e x z e y x .(4.12)
,
-
152
2 2
2 2
2 2
( )
+ ( )
+ ( )
x x i i i y i i i z i i ii i i
y y i i i z i i i x i i ii i i
z z i i i x i i i y i i ii i i
L e m y z m x y m x z
e m z x m y z m y x
e m x y m z x m z y
(4.13)
2 2( )i i i x xxi
m y z I I x.
i i i xyi
m x y I
