Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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E ∝ 1/l
2
0 l
−mv
mv
p
x
2
˙
l ∆t
2l
v
≪ ∆t ≪
l
˙
l
.
∆t
v∆ t/(2l)
∆v = −v
˙
l
∆t
l
.
∆v ∆t
∆v/∆t = dv/dt
vl = const
t
vl
в)
t
v
2l/v
2
˙
l
б)
t
l
а)
l, v, lv
vl
l(t) v(t) vl
vl
hvli
∼
˙
l/v hvli
d
dt
hvli ∼
˙
l
2
.
l
vl
I
˙
I = −
˙
λ
∂Λ(I, w, λ)
∂w
. (43.11)
˙
λ
∂Λ(I, w, λ)/∂w
˙
λ
I = const, w = ωt + ω
0
, λ = const .
∂Λ(I, w, λ)
∂w
=
1
ω
dΛ
dt
dI
dt
= −
˙
λ
ω
dΛ
dt
. (43.12)
¨
λT ≪
˙
λ
˙
λ = const
I(t + T ) − I(t) = −
˙
λ
ω
[Λ(t + T ) − Λ ( t)] = 0 , (43.13)
Λ
I(t)
˙
λT Λ
I(t)
I = mvl/π
“
m
M ≫ m
h
“
z
B
ϕ
= 0, B
z
= B
z
(z), B
r
= −
r
2
B
′
z
(z)
B
z
(z) = B
0
1 +
z
2
a
2
.
2s
q, p
s
I
i
(q
1
, . . . , q
s
, p
1
, . . . , p
s
) = const , i = 1, 2, . . . , s , (44.1)
s
w
1
, w
2
q
i
, p
i
w
i
0 ≤ w
i
≤ 2π w
1
, w
2
“
ω
1
/ω
2
q, p
s “
w
1
w
2
2π
2π
w
1
, w
2
ω
i
/ω
j
w
i
, w
j
“ ∂H/∂t = 0
H(q, p) = E
2s−1
“
ϕ
0
∼ 10
−8
l
a l ≫ a
OO
′
∼ lϕ
0
AA
′
∼
1
2
OO
′
∼ lϕ
0
, α ∼
AA
′
a
, ϕ
1
∼ α ∼
l
a
ϕ
0
.
“ “
a
l
ϕ
0
v
0
v
′
0
O
O
′
v
1
v
′
1
A
A
′
α
ϕ
1
“
“ α
ϕ
1
∼ α
ϕ
1
∼
l
a
ϕ
0
≫ ϕ
0
,
k
ϕ
k
∼
l
a
k
ϕ
0
.
l/a ∼ 10 k ∼ 8÷10
ϕ
1
∼ 1
“
∼ 100
“
“
“
“
“
“