Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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dM
dt
= [l, mg] . (47.20 )
M l
M = hMi + δM l = hli+ δl
h. . .i
hl i =
M
M
l cos α
M
, (47.21)
α
M
M
dhMi
dt
= [hl i, mg] = [Ω , hMi] , Ω = −
ml cos α
M
M
g .
(47.22)
Ω
Ω Ω
Ω
Ω
∼
mgl
M
2
/I
′
1
∼
mgl
T
≪ 1 . (47.23)
Ω
ω
O
Ω
Ω
O
θ
Ω
a = 6,4
M
m
R = 380
Ω
α
ω
x
1
x
2
x
3
K
XY Z K
0
K
0
K
ϕ Z x
1
N
ϕ
ψ
˙
ψ
˙
ϕ
θ
˙
θ
N
Z
Y
X
x
3
x
2
x
1
θ
x
3
ψ
ϕ, θ, ψ
˙
ϕ
˙
θ
˙
ψ Z
N x
3
Ω =
˙
ϕ +
˙
θ +
˙
ψ
K
˙
ϕ θ (π/2) −ψ
x
2
θ ϕ
x
3
˙
θ π/2 (−ψ)
˙
ψ
x
3
Ω
1
= ˙ϕ sin θ sin ψ +
˙
θ cos ψ ,
Ω
2
= ˙ϕ sin θ cos ψ −
˙
θ sin ψ , (48.1)
Ω
3
= ˙ϕ cos θ +
˙
ψ .
K
T =
1
2
I
1
Ω
2
1
+ I
2
Ω
2
2
+ I
3
Ω
2
3
, ( 48.2)
x
1
x
2
x
3
I
1
= I
2
T =
1
2
h
I
1
(
˙
θ
2
+ ˙ϕ
2
sin
2
θ) + I
3
( ˙ϕ cos θ +
˙
ψ)
2
i
. (48.3)
L L = T
p
ψ
=
∂L
∂
˙
ψ
= I
3
Ω
3
(48.4)
M
3
x
3
p
ϕ
=
∂L
∂ ˙ϕ
= I
1
Ω
1
sin θ sin ψ + I
2
Ω
2
sin θ cos ψ + I
3
Ω
3
cos θ ( 4 8 .5 )
M
Z
OZ
p
θ
=
∂L
∂
˙
θ
= I
1
Ω
1
cos ψ − I
2
Ω
2
sin ψ (48.6)
ϕ p
ϕ
= M
Z
p
ψ
= M
3
ψ
xy
A(x
1
, y
1
) B( x
2
, y
2
)
y(x)
l =
B
Z
A
dl =
x
2
Z
x
1
p
1 + (dy/dx)
2
dx, (dl)
2
= (dx)
2
+(dy)
2
(A.1)
y(x)
y(x
1
) = y
1
, y(x
2
) = y
2
.
y(x) = y
1
+
y
2
− y
1
x
2
− x
1
(x − x
1
) . (A.2)
˜y(x)
A(x
1
, y
1
) B(x
2
, y
2
)
˜y(x
1
) = y
1
, ˜y(x
2
) = y
2
y(x)
J =
x
2
Z
x
1
f(y, y
′
, x)dx, y
′
=
dy(x)
dx
(A.3)
y(x)
˜y(x) = y(x) + εh(x), h(x
1
) = h(x
2
) = 0 , (A.4)
ε h(x)
y
1
y
2
x
1
x
2
y(x)
˜y(x) = y(x) + δy(x)
εh(x)
y(x)
δy(x) ≡ εh( x).
˜y y
J(ε) =
x
2
Z
x
1
f(y + εh, y
′
+ εh
′
, x)dx , h
′
=
dh(x)
dx
,
ε
y(x)
J(ε) ε = 0 J(ε)
dJ(ε)
dε
|
ε=0
=
x
2
Z
x
1
∂f
∂y
h +
∂f
∂y
′
h
′
dx = 0.
h(x
1
) = h(x
2
) = 0
x
2
Z
x
1
∂f
∂y
−
d
dx
∂f
∂y
′
h(x)dx +
∂f
∂y
′
h(x)|
x
2
x
1
=
=
x
2
Z
x
1
∂f
∂y
−
d
dx
∂f
∂y
′
h(x) dx = 0. (A.5)
g(x)
x
2
Z
x
1
g(x) h(x)dx = 0
h(x)
h(x
1
) = h(x
2
) = 0 g(x) = 0
x ∈ (x
1
, x
2
)
y(x)
∂f
∂y
−
d
dx
∂f
∂y
′
= 0 ( A.6)
d
dx
y
′
p
1 + y
′2
= 0,
y
′
= const y(x)
y(x
1
) = y
1
, y(x
2
) = y
2
J y(x)
δJ
δy(x)
≡
∂f
∂y
−
d
dx
∂f
∂y
′
. (A.7)
J
δJ
δJ ≡
x
2
Z
x
1
δJ
δy(x)
δy(x) dx . (A.8)
J =
x
2
Z
x
1
f(y
1
, ..., y
s
; y
′
1
, ..., y
′
s
, x) dx, (A.9)
y
i
(x)
δJ
δy
i
(x)
≡
∂f
∂y
i
−
d
dx
∂f
∂y
′
i
= 0, i = 1, 2, ..., s. (A.10)