Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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dW
dt
= 0 ,
W
x(t) x(t + T )
ω x
1
(t)
x
2
(t)
x
1
(t + T ) = a
11
x
1
(t) + a
12
x
2
(t),
x
2
(t + T ) = a
21
x
1
(t) + a
22
x
2
(t).
a
ij
x
1
(t) x
2
(t)
T
x
1
(t + T ) = µ
1
x
1
(t) , x
2
(t + T ) = µ
2
x
2
(t) , (C.3)
µ
1,2
a
ij
W (x
1
(t + T ), x
2
(t + T )) = µ
1
µ
2
W (x
1
(t), x
2
(t)) .
µ
1
µ
2
= 1. (C.4)
a
ij
µ
±
= d ±
p
d
2
− 1 , d =
1
2
(a
11
+ a
22
) . (C.5)
d
2
> 1 µ
+
µ
−
µ
1
µ
1
> 1 d > 1
µ
1
= µ
+
µ
1
< −1 d < −1
µ
1
= µ
−
x
1
(t)
T
x
2
(t)
d
2
< 1 µ
1
µ
2
(µ
2
= µ
∗
1
)
(|µ
1
| = |µ
2
| = 1)
x
1,2
(t)
x
2
(t) = x
∗
1
(t)
x
1
(t) + x
2
(t) [x
1
(t) −x
2
(t)]/i
x
i
(t)
T
µ
i
x
i
(t)/µ
t/T
i
x
1
(t) = µ
t/T
1
Π
1
(t), x
2
(t) = µ
t/T
2
Π
2
(t) , (C.6)
Π
1
(t) Π
2
(t) µ
1
µ
2
¨x + 2λ ˙x + ω
2
(t) x = 0 . (C.7)
x = e
−λt
˜x(t) , (C.8)
˜x(t)
ω
2
(t) = ω
2
0
(1 + h cos γt) , (C.9)
γ = 2π/T h ≪ 1
¨x + ω
2
0
(1 + h cos γt) x = 0 , (C.10)
h γ
¨x + 2λ ˙x + ω
2
0
(1 + h cos γt) x = 0 , (C.10a) .
µ
1
γ h
µ
1
(γ, h) = −1 µ
1
(γ, h) = 1
(γ, h)
h h = 0 h = 0
ω = ω
0
x
1
(t) = a e
iω
0
t
.
ω T = 2π/γ
x
1
(t) = a e
iω
0
t
= a e
i(ω
0
−γ)t
e
iγt
= e
i(ω
0
−γ)t
Π
1
(t) ,
µ
1
= exp
i(ω
0
− γ)
2π
γ
.
ω
0
γ µ
1
= ±1
ω
0
γ
=
n
2
,
n ≥ 1 µ
1
= −1 n
µ
1
= 1 n h ≪ 1
γ
γ = 2ω
0
n = 1
γ = 2ω
0
+ ǫ ,
ǫ
µ
1
(−1)
µ
1
= − e
sT
= e
sT −iπ
,
s
Π(t)
Π(t) =
+∞
X
n=−∞
A
n
e
inγt
A
n
x(t) = e
st
+∞
X
n=−∞
A
n
e
i (2n−1)(ω
0
+ǫ/2) t
. (C.11)
cos γt
A
n
n
ω
2
0
+ [s + i(2n − 1)(ω
0
+ ǫ/2)]
2
o
A
n
=
= −
h
2
ω
2
0
(A
n−1
+ A
n+1
) . (C.12)
s
(−∞ < n < +∞)
h
h → 0
ω
0
x(t) → a cos (ω
0
t + ϕ) = A
+
e
iω
0
t
+ A
−
e
−iω
0
t
, (C.13)
A
±
=
1
2
a e
±iϕ
.
s = ǫ = 0 A
0
A
1
h
A
−1
= (h/16) A
0
A
2
= (h/16) A
1
2 (ǫ + 2is) A
0
− hω
0
A
1
= 0,
2 (ǫ − 2is) A
1
− hω
0
A
0
= 0.
(C.14)
s =
1
4
q
(hω
0
)
2
− 4ǫ
2
.
ǫ
(s = 0)
γ = 2ω
0
±
h
2
ω
0
.
h = 0
h
2ω
0
h
p1
γ
2ω
0
h
a
exp(−λt)
s = −λ +
1
4
q
(hω
0
)
2
− 4ǫ
2
.
h > h
p1
=
4λ
ω
0
s
γ = ω
0
n = 2 γ =
ω
0
+ ǫ µ
1
µ
1
= e
sT
x(t) = e
st
+∞
X
n=−∞
A
n
e
i n(ω
0
+ǫ) t
(C.15)
A
n
n
ω
2
0
+ [s + in(ω
0
+ ǫ)]
2
o
A
n
= −
h
2
ω
2
0
(A
n−1
+ A
n+1
) , (C.16)
(2n −
1) → n ǫ/2 → ǫ
h s = ǫ = 0
A
−1
A
1
h
A
±2
=
h
16
A
±1
, A
0
=
h
2
(A
−1
+ A
1
) . (C.17)
(ǫ + is) A
−1
= 0 , (ǫ − is) A
1
= 0 ,
s = ǫ = 0
h
h
4 (ǫ + is) A
−1
= hω
0
(A
−2
+ A
0
) ,
4 (ǫ − is) A
1
= hω
0
(A
2
+ A
0
) .
(C.18)
A
−2
, A
0
, A
2
A
−1
A
1
A
−1
A
1
s =
1
8
s
h
4
ω
2
0
−
8ǫ +
2h
2
3
ω
0
2
.
h
ω
0
h
p2
γγ
min
ω
0
γ = ω
0
−
5h
2
24
ω
0
,
γ = ω
0
+
h
2
24
ω
0
.
h = 0
h
2
s = −λ +
1
8
s
h
4
ω
2
0
−
8ǫ +
2h
2
3
ω
0
2
,
h = h(γ)
γ
min
= ω
0
−
h
2
12
ω
0
h
h
p2
=
r
8λ
ω
0
.
h > h
p2
(λ ≪ ω
0
)
h
p2
/h
p1
≫ 1
x = x(ξ)
Z
f(x) dx
f = f(x(ξ)) , dx =
dx
dξ
dξ .
x = x(ξ, η), y = y(ξ, η)
I =
Z
f(x, y) dxdy ;
dx dy (D.1)
D dξ dη,
D =
∂(x, y)
∂(ξ, η)
xy → ξη
dx =
∂x
∂ξ
dξ +
∂x
∂η
dη , dy =
∂y
∂ξ
dξ +
∂y
∂η
dη . (D.2)
dxdy
dx ∧ dy,
dy ∧ dy = −dx ∧ dy . (D.3)
dx ∧ dx = 0.