Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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cos (γt + ϕ) = Re
h
e
i(γt+ϕ)
i
.
x = Re
h
Be
i(γt+ϕ)
i
,
B = be
iδ
=
f
m(ω
2
0
− γ
2
+ 2iλγ)
.
B
γ
γ = ω
0
δ 0 > δ > −π
b δ
b =
f
m
p
(ω
2
0
− γ
2
)
2
+ 4λ
2
γ
2
, ctg δ =
γ
2
− ω
2
0
2λγ
. (22.4)
f
b
γ
m
=
q
ω
2
0
− 2λ
2
. (22.5)
b
max
= b(γ
m
) =
f
2mλ
p
ω
2
0
− λ
2
(22.6)
b δ
γ λ
λ δ
b
f/k
ω
0
γ
δ
0
−π/2
−π
ω
0
γ
b δ
γ
λ
γ < ω
0
δ (−π)
γ > ω
0
γ ω
0
x = b cos(γt + ϕ + δ) .
δ
λ ≪ ω
0
ǫ = γ − ω
0
(22.7)
γ
2
− ω
2
0
= (γ + ω
0
)(γ − ω
0
) ≈ 2ω
0
ǫ, 2λγ ≈ 2λω
0
, (22.8)
b =
f
2mω
0
√
ǫ
2
+ λ
2
, ctg δ =
ǫ
λ
. (22.9)
λ
b
max
=
f
2mω
0
λ
. (22.10)
ǫ = ±λ
δ
λ
|ǫ| ≫ λ
U
x = A cos (ωt + ϕ)
F
g
L(r, v) = L
0
+ L
g
, L
0
=
m
2
v
2
−U(r) , L
g
= v ·C(r) , (23.1)
C(r) r L
0
U(r) L
g
F
g
d
dt
∂L
∂v
−
∂L
∂r
= 0
m
¨
r = − ∇U(r) + F
g
, F
g
= [v, [∇, C(r)]] . (23.2)
F
g
v = 0
E =
m
2
v
2
+ U(r)
e
B(r)
C(r) =
e
c
A(r) ,
A(r)
F
g
=
e
c
[v, B(r)] .
Ω
C(r) = m [Ω, r] ,
F
g
= 2m[v, Ω] .
U U
U(x, y) =
1
2
k
x
x
2
+
1
2
k
y
y
2
,
z
B = (0 , 0, B) .
xy U(x, y)
x = y = 0 ω
B
= eB/(mc) k
B
= mω
2
B
> 0
m¨x +k
x
x − mω
B
˙y = 0,
m¨y +k
y
y + mω
B
˙x = 0.
x = R e(Ae
iωt
), y = Re(Be
iωt
) ,
A = ae
iϕ
(k
x
− mω
2
)A − imω
B
ωB = 0,
imω
B
ωA + (k
y
− mω
2
)B = 0
m
2
ω
4
− mω
2
(k
B
+ k
x
+ k
y
) + k
x
k
y
= 0.
ω
2
1,2
=
1
2m
k
B
+ k
x
+ k
y
±
q
(k
B
+ k
x
+ k
y
)
2
− 4k
x
k
y
k
x
> 0, k
y
> 0 (23.3)
x =
y = 0
k
x
< 0, k
y
< 0 (23.4a)
k
B
> − k
x
− k
y
+ 2
p
k
x
k
y
(23.4b)
x = y = 0
x = a
α
cos(ω
α
t + ϕ
α
) ,
y = b
α
sin(ω
α
t + ϕ
α
) ,
b
α
=
mω
B
ω
α
k
y
− mω
2
α
a
α
, α = 1 , 2 .
k
x
> k
y
> 0
x
y
x y
ω
2
1,2
x y
U
y
x
U
z =
x
2
2a
+
y
2
2b
,
z
Ω g = (0, 0, −g)
Ω
r v
L(r, v) =
m
2
(v + [Ω, r])
2
+ mgr =
=
m
2
( ˙x − Ωy)
2
+ ( ˙y + Ωx)
2
+ ˙z
2
− g
x
2
a
+
y
2
b
.
m ˙z
2
/2
k
x
m
→
g
a
− Ω
2
,
k
y
m
→
g
b
− Ω
2
, ω
B
→ 2Ω .
g
a
− Ω
2
g
b
− Ω
2
> 0 ,
g
a
− Ω
2
g
b
− Ω
2
< 0
a > b
g
a
< Ω
2
<
g
b
.
Ω
2
> g/b
U = −
m
2
Ω
2
−
g
a
x
2
−
m
2
Ω
2
−
g
b
y
2
◦