Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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ϕ
1
l
1
m
ϕ
2
l
2
m
k
ϕ
x
y
Q
1
Q
2
x = l
1
ϕ
1
y = l
2
ϕ
2
L(x, y, ˙x, ˙y) =
m
2
˙x
2
− ω
2
x
x
2
+ ˙y
2
− ω
2
y
y
2
+ 2αxy
,
ω
x
=
s
g
l
1
+
kl
2
2
ml
2
1
, ω
y
=
r
g
l
2
+
k
m
, α =
kl
2
ml
1
.
α = 0
ω
x
ω
y
l
1
≥ l
2
ω
x
≤ ω
y
α ≪ ω
2
x
Q
1,2
(t) = a
1,2
cos (ω
1,2
t + χ
1,2
)
ϕ xy
x = Q
1
cos ϕ − Q
2
sin ϕ , y = Q
1
sin ϕ + Q
2
cos ϕ ,
tg 2ϕ =
2α
ω
2
y
− ω
2
x
,
ω
2
1,2
=
1
2
ω
2
x
+ ω
2
y
∓
q
ω
2
y
− ω
2
x
2
+ 4α
2
,
ω
1
< ω
x
ω
2
> ω
y
ϕ α ≪ ω
2
x
α ≪ ω
2
y
− ω
2
x
. (20.9)
ϕ ≈ 0
x ≈ Q
1
, y ≈ Q
2
,
ω
1
≈ ω
x
ω
2
≈
ω
y
α ≪ ω
2
x
α ≫ ω
2
y
−ω
2
x
ϕ ≈ π/4
x ≈
Q
1
− Q
2
√
2
, y ≈
Q
1
+ Q
2
√
2
. (20.10)
l
1
= l
2
= l
x(0) = x
0
, y(0) = ˙x(0 ) = ˙y(0) = 0 , (20.11)
Q
1
(t) =
x
0
√
2
cos ω
1
t , Q
2
(t) = −
x
0
√
2
cos ω
2
t ,
ω
1
=
r
g
l
, ω
2
=
r
g
l
+
2k
m
,
ϕ
1
(t) =
x
0
l
cos εt cos ωt , ϕ
2
(t) =
x
0
l
sin εt sin ωt , (20.12)
ε =
1
2
(ω
2
− ω
1
) ≈
k
2m
s
l
g
≪ ω =
1
2
(ω
2
+ ω
1
) ≈
r
g
l
.
τ = π/(2ε)
2τ
k k
m
1
m
2
x
1
x
2
A B
AB
m
1
= m
2
m
1
≪ m
2
m
1
≫ m
2
k k
k
M
m m
1
32
k k
k
m
M m
1
32
k k
k
m
m m
1
32
M → m
x
1
(0) = −x
2
(0) = a, x
3
(0) = 0
δk
f = −kx
f(t)
∆U(x, t) = −xf(t)
L(x, ˙x, t) =
1
2
(m ˙x
2
− kx
2
) + xf(t) . (21.1)
¨x + ω
2
x = f(t)/m, ω =
p
k/m (21.2)
f(t)
x(0) = x
0
, ˙x(0) = v
0
x(t) = x
0
cos ωt +
v
0
ω
sin ωt +
t
Z
0
f(τ)
sin ω(t − τ)
ωm
dτ . (21.3)
f(t) = f cos(γt + ϕ)
x = b cos(γt + ϕ), b =
f
(ω
2
− γ
2
)m
. (21.4)
b γ
γ < ω
b
f/k
ω γ
b
γ
(b/f > 0) γ > ω
(b/f < 0) γ ≫ ω
b/f ≪ 1/k
b ≈ −
f
mγ
2
, γ ≫ ω . (21.4a)
γ → ω b → ∞
∆L =
X
i
x
i
F
i
(t).
F(t) =
F
1
(t)
F
2
(t)
F
s
(t)
,
L(x,
˙
x, t) =
1
2
(
˙
x, ˆm
˙
x) −
1
2
(x,
ˆ
k x) + x F(t) ; (21.1 a)
ˆm
¨
x +
ˆ
k x = F(t) . (21.2a)
x
Q
xF(t) =
X
α
A
(α)
F(t) · Q
α
L =
s
X
α=1
L
α
, L
α
=
1
2
(M
α
˙
Q
2
α
− K
α
Q
2
α
) + Q
α
f
α
(t) , (21.5)
M
α
K
α
f
α
(t) = A
(α)
F(t) . (2 1 .6 )
Q
α
¨
Q
α
+ ω
2
α
Q
α
=
f
α
(t)
M
α
, ω
α
=
r
K
α
M
α
. (21.2b)
F(t) = F cos(γt +
ϕ)
F
α
(t) = f
α
cos(γt + ϕ) , f
α
= A
(α)
F (21.7)
Q
α
= b
α
cos(γt + ϕ) , b
α
=
f
α
(ω
2
α
− γ
2
)M
α
. (21.3a)
x =
s
X
α=1
A
(α)
b
α
cos(γt + ϕ) =
=
s
X
α=1
A
(α)
(A
(α)
, F)
(ω
2
α
− γ
2
)
A
(α)
, ˆm A
(α)
cos(γt + ϕ) . (21.8)
A
(α)
f
α
= A
(α)
F α
F
F A
(α)
F A
(α)
= 0
A
(α)
b
α
cos(γt + ϕ)
γ → ω
α
F A
(α)
6= 0 γ → ω
α
x ≈ A
(α)
b
α
cos(γt + ϕ)
F
F = const · A
(1)
.
γ → ω
1
γ → ω
2
, γ → ω
3
, ..., γ → ω
s
)
x = B cos(γt + ϕ) B
(−γ
2
ˆm +
ˆ
k)B = F ,
x = (−γ
2
ˆm +
ˆ
k)
−1
F cos(γt + ϕ) . (21.9)
A a(t) =
a cos γt γ
x
f = −α ˙x α
m¨x = −α ˙x − kx
m
ω
2
0
=
k
m
, λ =
α
2m
.
ω
0
λ
¨x + 2λ ˙x + ω
2
0
x = 0 . (22.1a)
x = Re
e
rt
,
r
r
1,2
= − λ ±
q
λ
2
− ω
2
0
.
x = R e
C
1
e
r
1
t
+ C
2
e
r
2
t
. (22.2)
(λ > ω
0
)
x
λ = ω
0
x = (C
1
+ C
2
t) e
−λt
.
(λ < ω
0
)
r
1,2
= − λ ±iω , ω =
q
ω
2
0
− λ
2
,
x = ae
−λt
cos(ωt + ϕ) , (22.3)
a ϕ
λ ≪ ω
0
T = 2π/ω
e
−λt
e
−2λt
E(t) = E
0
e
−2λt
.
f cos ( γt + ϕ)
¨x + 2λ ˙x + ω
2
0
x =
f
m
cos (γt + ϕ) . ( 2 2 .1 b)