Яковенко Г.Н. Краткий курс теоретической механики
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M
O
A = B = 2C
O
M
O
=
X
i
[r
i
, F
i
] = 0. (33.1)
A ˙p + (C − B)qr = 0,
B ˙q + (A − C)pr = 0,
C ˙r + (B − A)pq = 0.
(33.2)
δA V
i
= ˙r
i
= [ωωω
ωω
, r
i
]
δA =
X
i
(F
i
, dr
i
) =
X
i
(F
i
, V
i
)dt =
X
i
(F
i
, [ωωω
ωω
, r
i
])dt =
=
X
i
(ωωω
ωω
, [r
i
, F
i
])dt = (ωωω
ωω
,
X
i
[r
i
, F
i
])dt = (ωωω
ωω
, M
O
)dt = 0
Ap
2
+ Bq
2
+ Cr
2
= 2T = const,
A
2
p
2
+ B
2
q
2
+ C
2
r
2
= K
2
O
= const.
(33.3)
M
O
= 0 A = B = C
K
O
= Aωωω
ωω
= const ωωω
ωω
= K
O
/A = const
p = const q = const r = const ωωω
ωω
M
O
= 0 A = B 6= C
e
3
e
3
{e
1
, e
2
}
ωωω
ωω
α
q
O
a
r
w
K
O
e
3
p +q
2 2
j
1
j
3
r = const. (33.4)
p
2
+ q
2
=
1
A
(2T − Cr
2
) = const. (33.5)
ω
2
= p
2
+ q
2
+ r
2
= const (33.6)
α
tgα =
p
p
2
+ q
2
r
= const. (33.7)
K
O
e
3
ωωω
ωω
K
O
¯
¯
¯
¯
¯
¯
0 0 1
p q r
Ap Bq Cr
¯
¯
¯
¯
¯
¯
= (B − A)pq
A=B
= 0, (33.8)
θ e
3
K
O
tgθ =
A
p
p
2
+ q
2
Cr
= const. (33.9)
e
3
ωωω
ωω
K
O
j
1
K
O
{e
3
, ωωω
ωω
K
O
, j
1
}
K
O
O
w
K
O
e
3
j
1
j
3
w
1
w
2
q
{e
3
, ωωω
ωω
K
O
, j
1
}
j
1
j
2
j
3
j
2
K
O
ωωω
ωω
2
= ωωω
ωω
ωωω
ωω
1
= ωωω
ωω
ωωω
ωω
1
ωωω
ωω
2
ωωω
ωω
ωωω
ωω
= ωωω
ωω
1
+ ωωω
ωω
1
ωωω
ωω
1
= ωωω
ωω
ωωω
ωω
2
= ωωω
ωω
θ i
3
K
O
i
1
i
2
i
3
n
j
2
ωωω
ωω
1
ωωω
ωω
2
θ
ω
1
= ˙ϕ ω
2
=
˙
ψ θ = θ
M
O
= 0 A = B 6= C
ωωω
ωω
p q r
Ap Bq Cr K
O
K
O
e
3
θ
ωωω
ωω
e
3
K
O
ωωω
ωω
1
e
3
ωωω
ωω
2
K
O
M
O
= 0 A < B < C
Ap
2
+ Bq
2
= 2T − Cr
2
,
A
2
p
2
+ B
2
q
2
= K
2
O
− C
2
r
2
.
(34.1)
p
2
q
2
∆ =
AB(B − A) 6= 0 p = f(r) q = h(r)
dr
dt
=
1
C
(A − B)f(r)h(r) = F (r)
r(t) p(t) = f(r(t)) q(t) = h(r(t))
p(t) q(t) r(t)
M
O
= 0
A < B < C O
O
O
P
r
P
w
Q
K
O
N
f(y) = Ay
2
1
+ By
2
2
+ Cy
2
3
− 1 = 0. (34.2)
ωωω
ωω
P
r
P
V
P
= [ωωω
ωω
, r
P
] = 0, r
P
= λωωω
ωω
, (34.3)
r
P
ωωω
ωω
y
1
= λp y
2
= λq y
3
= λr
Ay
2
1
+ By
2
2
+ Cy
2
3
− 1 = λ
2
(Ap
2
+ Bq
2
+ Cr
2
) − 1 = 2λ
2
T − 1 = 0
λ
2
=
1
2T
= const. (34.4)
N P
N = gradf(y) =
µ
∂f
∂y
1
,
∂f
∂y
2
,
∂f
∂y
3
¶
= 2(Ap, Bq, Cr) = 2K
O
= const, (34.5)
P
P
OQ O
OQ =
µ
r
P
,
K
O
K
O
¶
=
λ
K
O
(Ap
2
+ Bq
2
+ Cr
2
) =
2λT
K
O
= const.
O
M
O
= 0 A = B 6= C
O
A = B 6= C
ω
1
ω
2
θ ωωω
ωω
1
F
w
1
q
e
1
e
3
O
w
2
K
O
P
C
e
3
{ωωω
ωω
1
ωωω
ωω
2
} e
1
ωωω
ωω
1
ωωω
ωω
2
ωωω
ωω
= ωωω
ωω
1
+ ωωω
ωω
2
K
O
ωωω
ωω
1
= ω
1
e
3
, ωωω
ωω
2
= ω
2
(e
1
sin θ + e
3
cos θ),
ωωω
ωω
= ωωω
ωω
1
+ ωωω
ωω
2
= pe
1
+ qe
2
+ re
3
= ω
2
e
1
sin θ + (ω
1
+ ω
2
cos θ)e
3
,
K
O
= Ape
1
+ Bqe
2
+ Cre
3
= Aω
2
e
1
sin θ + C(ω
1
+ ω
2
cos θ)e
3
.
M
O
=
dK
O
dt
= V
F
= [ωωω
ωω
2
, K
O
] = −e
2
ω
2
{Cω
1
+ ω
2
(C − A) cos θ}sin θ. (35.1)
M
O
= [ωωω
ωω
2
, ωωω
ωω
1
]
½
C +
ω
2
ω
1
(C − A) cos θ
¾
. (35.2)
C P
OC = a
O
ωωω
ωω
2
O
M
O
= −mgae
2
sin θ
mga sin θ = ω
2
{Cω
1
+ ω
2
(C − A) cos θ}sin θ.
sin θ = 0 OC
ω
1
ω
2
sin θ 6= 0 ω
1
ω
2
θ
ω
2
2
(C − A) cos θ + Cω
1
ω
2
− mga = 0.
ω
1
À ω
2
M
O
= C[ωωω
ωω
2
, ωωω
ωω
1
].
C
mW
C
= R
R
C
−m
i
W
C
M
C
= −
X
i
[r
i
, m
i
W
C
] = −[
X
i
m
i
r
i
, W
C
] = −[mr
C
, W
C
] = 0,
r
i
r
C
C
C