Яковенко Г.Н. Краткий курс теоретической механики
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ωωω
ωω
∗
= pi
1
+ qi
2
+ ri
3
,
˙
Λ =
1
2
Λ ◦ωωω
ωω
∗
(29.4)
˙
λ
0
+
3
X
k=1
˙
λ
k
i
k
=
1
2
(λ
0
+
3
X
k=1
λ
k
i
k
) ◦ (pi
1
+ qi
2
+ ri
3
).
˙
λ
0
˙
λ
1
˙
λ
2
˙
λ
3
=
1
2
0 −p −q −r
p 0 r −q
q −r 0 p
r q −p 0
λ
0
λ
1
λ
2
λ
3
. (29.5)
e
e
3
e
1
O
e
2
r
i
h
i
a
w
K
O
F
m
i
O e
1
e
2
e
3
O e
e α
k
e =
3
P
k=1
α
k
e
k
e
1
e
2
e
3
α
k
= (e, e
k
) = cos{[e, e
k
}
3
X
k=1
α
2
k
= (e, e) = 1. (30.1)
I
e
e
I
e
=
X
i
m
i
h
2
i
, (30.2)
m
i
h
i
e
r
i
=
3
P
k=1
x
ik
e
k
h
2
i
= r
2
i
− (r
i
, e)
2
1 − α
2
1
= α
2
2
+ α
2
3
1 − α
2
2
= α
2
1
+ α
2
3
1 − α
2
3
= α
2
1
+ α
2
2
I
e
=
X
i
m
i
h
2
i
=
X
i
m
i
©
r
2
i
− (r
i
, e)
2
ª
=
=
X
i
m
i
©¡
x
2
i1
+ x
2
i2
+ x
2
i3
¢
− (x
i1
α
1
+ x
i2
α
2
+ x
i3
α
3
)
2
ª
=
= α
2
1
X
i
m
i
¡
x
2
i2
+ x
2
i3
¢
+ α
2
2
X
i
m
i
¡
x
2
i1
+ x
2
i3
¢
+ α
2
3
X
i
m
i
¡
x
2
i1
+ x
2
i2
¢
−
−2α
1
α
2
X
i
m
i
x
i1
x
i2
− 2α
1
α
3
X
i
m
i
x
i1
x
i3
− 2α
2
α
3
X
i
m
i
x
i2
x
i3
.
e
k
I
1
=
X
i
m
i
¡
x
2
i2
+ x
2
i3
¢
, I
2
=
X
i
m
i
¡
x
2
i1
+ x
2
i3
¢
, I
3
=
X
i
m
i
¡
x
2
i1
+ x
2
i2
¢
, (30.3)
I
12
= I
21
=
P
i
m
i
x
i1
x
i2
, I
13
= I
31
=
P
i
m
i
x
i1
x
i3
,
I
23
= I
32
=
P
i
m
i
x
i2
x
i3
,
(30.4)
e =
3
P
k=1
α
k
e
k
I
e
=
3
X
k=1
I
k
α
2
k
− 2
X
k<l
I
kl
α
k
α
l
. (30.5)
˘α =
α
1
α
2
α
3
(30.6)
˘α
T
=
¡
α
1
α
2
α
3
¢
, ˘α
T
˘α = α
2
1
+ α
2
2
+ α
2
3
, ˘α˘α
T
=
α
2
1
α
1
α
2
α
1
α
3
α
2
α
1
α
2
2
α
2
α
3
α
3
α
1
α
3
α
2
α
2
3
.
˘
I =
I
1
−I
12
−I
13
−I
21
I
2
−I
23
−I
31
−I
32
I
3
(30.7)
e =
3
P
k=1
α
k
e
k
I
e
= ˘α
T
˘
I ˘α. (30.8)
O
e a = e/
√
I I
e =
3
P
k=1
α
k
e
k
a =
e
√
I
=
3
X
k=1
α
k
√
I
e
k
=
3
X
k=1
y
k
e
k
, y
k
=
α
k
√
I
. (30.9)
α
k
y
k
I
y
k
a
1 =
3
X
k=1
I
k
y
2
k
− 2
X
k<l
I
kl
y
k
y
l
, (30.10)
a
%
é
1
&
I
O
I > 0 1/
√
I < ∞
O
1/
√
I
O
O
I
O
1/
√
I
O
e
1
e
2
e
3
3
X
k=1
I
k
y
2
k
= 1, (30.11)
e
1
e
2
e
3
α
k
e
I
e
=
3
X
k=1
I
k
α
2
k
, (30.12)
e
1
e
2
e
3
O
O
e
3
O
I
13
= 0 I
23
= 0 e
3
O
O
I
1
= I
2
O
C
I I
C
C
I = I
C
+ md
2
, (30.13)
m d
¤ C e
1
e
2
e
3
e
3
C e
2
h
i
r
i
=
3
P
k=1
x
ik
e
k
h
2
i
= x
2
i1
+ (x
i2
− d)
2
I =
X
i
m
i
h
2
i
=
X
i
m
i
{x
2
i1
+ (x
i2
− d)
2
} =
=
X
i
m
i
(
x
2
i1
+
x
2
i2
) +
X
i
m
i
d
2
−
2
X
i
m
i
x
i2
d
=
I
C
+
md
2
.
P
i
m
i
x
i2
= mx
C2
= 0 ¥
O
O
O
O
r
i
ωωω
ωω
V
i
= [ωωω
ωω
, r
i
] V
i
= h
i
ω
e =
3
P
k=1
α
k
e
k
ωωω
ωω
ωωω
ωω
= ωe = ω
3
X
k=1
α
k
e
k
=
3
X
k=1
ω
k
e
k
, ω
k
= ωα
k
. (31.1)
T
T =
1
2
X
i
m
i
V
2
i
=
1
2
Ã
X
i
m
i
h
2
i
!
ω
2
=
1
2
I
ω
ω
2
=
1
2
Ã
3
X
k=1
I
k
α
2
k
− 2
X
k<l
I
kl
α
k
α
l
!
ω
2
,
T =
1
2
3
X
k=1
I
k
ω
2
k
−
X
k<l
I
kl
ω
k
ω
l
, (31.2)
I
k
I
kl
e
1
e
2
e
3
ω
1
ω
2
ω
3
ωωω
ωω
˘ω =
ω
1
ω
2
ω
3
(31.3)
T =
1
2
˘ω
T
˘
I ˘ω. (31.4)
e
1
e
2
e
3
I
1
= A, I
2
= B, I
3
= C (31.5)
ω
1
= p, ω
2
= q, ω
3
= r. (31.6)
T =
1
2
¡
Ap
2
+ Bq
2
+ Cr
2
¢
. (31.7)
V
i
= [ωωω
ωω
, r
i
]
K
O
=
X
i
[r
i
, m
i
V
i
] =
X
i
m
i
[r
i
, [ωωω
ωω
, r
i
]] =
X
i
m
i
©
ωωω
ωω
(r
i
, r
i
) − r
i
(r
i
, ωωω
ωω
)
ª
.
ωωω
ωω
=
3
P
k=1
ω
k
e
k
r
i
=
3
P
k=1
x
ik
e
k
K
O
e
1
K
1
= (K
O
, e
1
) =
X
i
m
i
©
ω
1
(x
2
i1
+ x
2
i2
+ x
2
i3
) − x
i1
(x
i1
ω
1
+ x
i2
ω
2
+ x
i3
ω
3
)
ª
=
= ω
1
X
i
m
i
(x
2
i2
+ x
2
i3
) − ω
2
X
i
m
i
x
i1
x
i2
− ω
3
X
i
m
i
x
i1
x
i3
= I
1
ω
1
− I
12
ω
2
− I
13
ω
3
.
K
j
K
O
=
3
P
j=1
K
j
e
j
K
1
= I
1
ω
1
− I
12
ω
2
− I
13
ω
3
,
K
2
= I
2
ω
2
− I
21
ω
1
− I
23
ω
3
,
K
3
= I
3
ω
3
− I
31
ω
1
− I
32
ω
2
.
(31.8)
˘
K
O
=
K
1
K
2
K
3
, (31.9)
˘
K
O
=
˘
I ˘ω. (31.10)
K
j
=
∂T
∂ω
j
.
e
1
e
2
e
3
K
j
K
O
=
3
P
j=1
K
j
e
j
K
1
= Ap, K
2
= Bq, K
3
= Cr. (31.11)
ωωω
ωω
K
O
[ωωω
ωω
, K
O
] =
¯
¯
¯
¯
¯
¯
e
1
e
2
e
3
p q r
Ap Bq Cr
¯
¯
¯
¯
¯
¯
= (C − B)qre
1
+ (A − C)pre
2
+ (B − A)pqe
3
. (31.12)
{ωωω
ωω
kK
O
} ⇔ {ωωω
ωω
∈ q = 0, r = 0}. (31.13)
O
O e
1
e
2
e
3
V
F
=
dK
O
dt
= M
O
=
3
X
k=1
M
k
e
k
. (32.1)
V
F
F K
O
F e
1
e
2
e
3
F
V
F
= A ˙pe
1
+ B ˙qe
2
+ C ˙re
3
. (32.2)
F
V
F
= [ωωω
ωω
, K
O
],
V
F
= [ωωω
ωω
, K
O
] = (C − B)qre
1
+ (A − C)pre
2
+ (B − A)pqe
3
. (32.3)
V
F
= V
F
+ V
F
A ˙p + (C − B)qr = M
1
,
B ˙q + (A − C)pr = M
2
,
C ˙r + (B − A)pq = M
3
.
(32.4)
M
k
M
O
A ˙p + (C − B)qr = M
1
(t, ψ, θ, ϕ, p, q, r),
B ˙q + (A − C)pr = M
2
(t, ψ, θ, ϕ, p, q, r),
C ˙r + (B − A)pq = M
3
(t, ψ, θ, ϕ, p, q, r),
˙
ψ = (p sin ϕ + q cos ϕ)
1
sin θ
,
˙
θ = p cos ϕ − q sin ϕ,
˙ϕ = (p sin ϕ + q cos ϕ)ctgθ.
(32.5)