Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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γ → ω
α
ω
α
ϕ →
ϕ
α
= πα/(N + 1)
sin(N + 1) ϕ →
sin(N +1)ϕ
α
= 0
γ > 2ω
0
y
n
= (−1)
n
A e
nψ
+ B e
−nψ
cos(γt + χ) , (28.5)
γ
2
= 4ω
2
0
ch
2
ψ
2
. (28.6)
y
0
= 0 A = −B
y
n
= (−1)
n
2A sh nψ cos( γt + χ) ,
(−1)
N+ 1
2A sh(N + 1)ψ = b
y
n
= (−1)
N+ 1−n
b
sh nψ
sh(N + 1)ψ
cos(γt + χ) . (28.7 )
γ ≫ 2ω
0
γ ≈ ω
0
e
−ψ
y
n
=
b
(−γ
2
/ω
2
0
)
N+ 1−n
cos(γt + χ) , (28.8)
a
γ ≫ 2ω
0
γ ≫
ω
α
(N − 1)
(N − 1)
N
L(x, ˙x) =
m ˙x
2
2
−
kx
2
2
−
mαx
3
3
−
mβx
4
4
,
α β
¨x + ω
2
0
x = − αx
2
− βx
3
, (29.1)
ω
2
0
= k/m
T = 2π/ω
x(t) =
∞
X
n=0
a
n
cos nωt . (29.2)
a
n
ω
t = 0 x
a
1
≡ a ,
δω = ω − ω
0
a
a
n
ω
a
x a
a
x
(1)
= a cos ωt
ω ω
0
x
(1)
cos
2
ωt =
1
2
+
1
2
cos 2ωt , cos
3
ωt =
3
4
cos ωt +
1
4
cos 3ωt .
¨x + ω
2
0
x = −
1
2
αa
2
(1 + cos 2ωt) −
1
4
βa
3
(3 cos ωt + cos 3ωt) .
x
(2)
(t) = a
(2)
0
+ a cos ωt + a
(2)
2
cos 2ωt + a
(2)
3
cos 3ωt .
ω
2
0
a
(2)
0
= −
αa
2
2
, (29.3)
(ω
2
− ω
2
0
)a =
3
4
βa
3
, (29.4)
(4ω
2
− ω
2
0
)a
(2)
2
=
1
2
αa
2
, (29.5)
(9ω
2
− ω
2
0
)a
(2)
3
=
1
4
βa
3
. (29.6)
ω
ω
2
− ω
2
0
= 2ω
0
(ω − ω
0
) ,
δω
(2)
= ω − ω
0
=
3βa
2
8ω
0
.
ω
0
ω
a
(2)
0
= −
αa
2
2ω
2
0
, a
(2)
2
=
αa
2
6ω
2
0
, a
(2)
3
=
βa
3
32ω
2
0
. (29.7)
a
a
a
a
a
(2)
0
a cos ωt a
(2)
2
a cos 2ωt cos ωt a
3
δω
(3)
= −
5α
2
a
2
12ω
3
0
a
(3)
3
=
α
2
a
3
48ω
4
0
.
a
a
x(t) = a cos ωt −
αa
2
6ω
2
0
(3 − cos 2ωt) +
a
3
16ω
2
0
α
2
3ω
2
0
+
β
2
cos 3ωt ,
δω =
3β
8ω
0
−
5α
2
12ω
3
0
a
2
. (29.8)
a ω
0
αa
ω
2
0
≪ 1,
βa
2
ω
2
0
≪ 1 . (29.9)
ω
1
, . . . , ω
α
, . . . , ω
β
, . . . , ω
s
|ω
α
±ω
β
|
l
k
m
y
x
m
k g
l
0
l = l
0
+ (mg/k)
x
y
L =
m
2
˙x
2
+ ˙y
2
−
k
2
h
p
(l − y)
2
+ x
2
− l
0
i
2
− mgy
L =
m
2
˙x
2
− ω
2
x
x
2
+ ˙y
2
− ω
2
y
y
2
+ 2αx
2
y
, (29.10 )
ω
x
=
r
g
l
, ω
y
=
r
k
m
, α =
kl
0
2ml
2
.
¨x + ω
2
x
x = 2αxy ,
¨y + ω
2
y
y = αx
2
x = x
(1)
+ x
(2)
+ . . . , y = y
(1)
+ y
(2)
+ . . . .
ω
x
ω
y
x
(1)
= a cos (ω
x
t + ϕ
x
) , y
(1)
= b cos (ω
y
t + ϕ
y
) . (29.11)
¨x
(2)
+ω
2
x
x
(2)
= 2α x
(1)
y
(1)
= α ab [cos (ω
+
t + ϕ
+
) + cos (ω
−
t + ϕ
−
)] ,
ω
±
= ω
y
± ω
x
, ϕ
±
= ϕ
y
± ϕ
x
.
ω
±
x
(2)
=
α ab
2ω
y
(2ω
x
+ ω
y
)
cos (ω
+
t + ϕ
+
)+
+
α ab
2ω
y
(2ω
x
− ω
y
)
cos (ω
−
t + ϕ
−
) . (29.12)
y
(2)
=
α a
2
2ω
2
y
−
α a
2
2(4ω
2
x
− ω
2
y
)
cos (2ω
x
t + 2ϕ
x
) , (29.13)
y
2ω
x
ω
y
2ω
x
ω
y
= 2ω
x
x y
2
f(t) = f cos γt
f = −2mλ ˙x
¨x + ω
2
0
x =
f
m
cos γt − αx
2
− βx
3
− 2λ ˙x . (30.1 )
x =
b cos(γt + δ)
x = b
0
+ b cos(γt + δ) + b
2
cos 2(γt + δ) + b
3
cos 3(γt + δ) + . . . .
(30.2)
a
γ f
b
ω
0
γ b
δ
−γ
2
+ ω
2
b cos(γt + δ) − 2λγb sin(γt + δ) =
f
m
cos γt , (30.3)