Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
Подождите немного. Документ загружается.
“
XY Z
“ x = x
1
, y =
x
2
, z = x
3
O
“
R O
XY Z
xyz XY Z
O
r
a
R
Y
z
Z
X
y
x
m
a
XY Z “
xyz “
N
a = 1, 2, . . . , N
m
a
r
a
xyz
R + r
a
.
x
a
, y
a
, z
a
r
a
xyz
r
a
R r
a
v
a
O
v
a
=
˙
R ≡ V . (45.1)
O
Ω
n
Ω = Ω · n .
Ω = Ω(t)
Ω
n a
v
a
= [Ω, r
a
] . (45.2)
e
i
x
1
x
2
x
3
e
i
e
k
= δ
ik
, i, k = 1, 2, 3 .
r
a
r
a
= x
a
e
1
+ y
a
e
2
+ z
a
e
3
. ( 45.3)
e
i
Ω
de
i
dt
= [Ω, e
i
] . (45.4)
x
a
, y
a
, z
a
r
a
A = A(t)
A = A
1
e
1
+ A
2
e
2
+ A
3
e
3
(45.5)
A
1
, A
2
, A
3
dA
dt
=
dA
1
dt
e
1
+
dA
2
dt
e
2
+
dA
3
dt
e
3
+ [Ω, A] . (45.6a)
e
i
dA
dt
i
≡ e
i
·
dA
dt
=
dA
i
dt
+
3
X
j,k=1
e
ijk
Ω
j
A
k
, i = 1, 2, 3 . ( 45.6b)
e
ijk
a
v
a
= V + [Ω, r
a
] . (45.7)
xyz O
Ω O
O
′
B
O r
′
a
r
a
r
a
= r
′
a
+ B . (45.8)
O
′
V
′
= V + [Ω, B] . (45.9)
v
a
v
a
= V
′
+ [Ω
′
, r
′
a
] (45.10)
V
′
+ [Ω
′
, r
′
a
] = V + [Ω, B] + [Ω, r
′
a
] ,
r
′
a
Ω
′
= Ω ,
O
V V
k
V
⊥
Ω(t)
V
′
V
′
k
= V
k
, V
′
⊥
= V
⊥
+ [Ω, B] .
V
k
= 0
O
′
V
′
O
′
P =
N
X
a=1
m
a
v
a
. (46.1a)
m =
X
a
m
a
, r =
P
a
m
a
r
a
m
, (46.2)
P = mV + m [Ω, r ] . (46.1 b)
xyz
r = 0
m
V = V
P = mV . (46.3)
M =
N
X
a=1
m
a
[R + r
a
, V + [ Ω, r
a
]] = m[ R, V]+
+
X
a
m
a
[r
a
, [Ω, r
a
]] + m[r , V] + m [R, [Ω, r ]] . (46.4)
xyz
m R
O Ω
M = m[R, V ] +
X
a
m
a
[r
a
, [ Ω, r
a
]] . (46.5)
O
M =
X
a
m
a
[r
a
, [Ω, r
a
]] . (46.6)
Ω
j
[r
a
, [Ω, r
a
]] = Ω r
2
a
− (Ω r
a
) r
a
,
x
1
x
2
x
3
[r
a
, [Ω, r
a
]] · e
i
=
3
X
k=1
r
2
a
δ
ik
− (r
a
)
i
(r
a
)
k
Ω
k
,
M
i
=
3
X
k=1
I
ik
Ω
k
, (46.7)
I
ik
I
ik
=
N
X
a=1
m
a
r
2
a
δ
ik
− (r
a
)
i
(r
a
)
k
, i, k = 1, 2, 3 . (46.8)
I
ik
M
Ω
T =
1
2
N
X
a=1
m
a
v
2
a
.
T =
1
2
mV
2
+
1
2
X
a
m
a
[Ω, r
a
]
2
+ m Ω[r , V] .
xyz
m
V
T =
1
2
mV
2
+
1
2
X
a
m
a
[Ω, r
a
]
2
. (46.9)
O
T =
1
2
X
a
m
a
[Ω, r
a
]
2
. (46.10)
T =
1
2
mV
2
=
1
2
PV . (46.11)
T =
1
2
MΩ . (46.12)
(Ω [r
a
, [Ω, r
a
]]) = [Ω, r
a
]
2
.
Ω
j
T =
1
2
3
X
i,k=1
I
ik
Ω
i
Ω
k
, (46.13)
I
ik
M Ω 90
o
T > 0
I
11
=
X
a
m
a
(y
2
a
+z
2
a
), I
22
=
X
a
m
a
(x
2
a
+z
2
a
), I
33
=
X
a
m
a
(x
2
a
+y
2
a
) ,
I
12
= I
21
= −
X
a
m
a
x
a
y
a
, I
13
= I
31
= −
X
a
m
a
x
a
z
a
,
I
23
= I
32
= −
X
a
m
a
y
a
z
a
.
I
ik
I
11
+ I
22
=
X
a
m
a
(x
a
+ y
2
a
+ 2z
2
a
) ≥
X
a
m
a
(x
2
a
+ y
2
a
) = I
33
.
xy
I
11
+ I
22
= I
33
.
n
ρ
a
a
n
I
n
=
X
a
m
a
ρ
2
a
.
I
11
I
22
I
33