Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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H(p, x) =
p
2
2m
+
1
2
mω
2
x
2
= E . (42.1)
x =
r
2I
mω
sin w, p =
√
2mωI cos w . (42.2)
{p, x}
I,w
= 1 ,
H
′
(I, w) = H(p(w, I), x(w, I)) = ωI
w
I =
E
ω
(42.3)
I =
˙w =
∂H
′
∂I
= ω (42.4)
w(t) = ω t + w
0
. (42.5)
xp
H(p, q) = E (42.6)
p
x
I
w
p, x I, w
p = p(q, E)
S
0
(q, E) =
q
Z
q
0
p(q, E) dq . (42.7)
I
I(E) =
1
2π
I
p(q, E) dq , (42.8)
p
q
q
0
q
S
0
(q, E)
2πI
2πI n
∆S
0
= 2πn I . (42.9)
E(I)
Φ(q, I) = S
0
(q, E(I)) . (42.10)
I
p =
∂Φ(q, I)
∂q
, w =
∂Φ(q, I)
∂I
. (42.12)
w
H
′
(I, w) = E(I) . (42.13)
˙
I = −
∂H
′
∂w
= −
∂E
∂w
= 0 , ˙w =
∂E
∂I
(42.14)
p(x, E) =
p
2mE −(mωx)
2
S
0
(x, E) =
x
Z
x
0
p(x, E) dx =
=
1
2
x
p
2mE − (mωx)
2
+
E
ω
arcsin
r
m
2E
ωx
+ 2πn
E
ω
+ const .
Φ(x, I) =
1
2
x
p
2mωI − (mωx)
2
+ I arcsi n
r
m
2ωI
ωx
+
+2πn I + const . (42.11)
I = const , w =
∂E
∂I
t + w
0
. (42.15)
∂E/∂I
∂I
∂E
=
1
2π
I
∂p(q, E)
∂E
dq . (42.16)
∂p(q, E)/∂E p =
p(q, E)
H (p(q, E), q) = E E
∂H
∂p
∂p
∂E
= 1 ,
∂p
∂E
=
1
∂H/∂p
=
1
˙q
, (42.17)
∂I
∂E
=
1
2π
I
dq
˙q
=
T
2π
=
1
ω
, (42.18)
T = 2π/ω ω
ω I
I = const , w = ω(I) t + w
0
. (42.19)
w
q
w 2π
q p w
∂H/∂t = 0
S
0
S
i
I
i
s
α
j
E
H
′
= E( I
1
, . . . , I
s
) , (4 2 .1 9)
q
i
, p
i
“ w
i
I
j
α
i
I
j
α
i
= α
i
(I
1
, . . . , I
s
) . (42.20)
I
i
I
i
= n
i
~
n
i
~
λ = λ(t)
H = H(p, q, λ)
E
w, I
λ
p = p(q, E, λ)
H(p, q, λ) = E (42.21)
λ
S
0
(q, E, λ)
Φ(q, I, λ)
H
′
(I, w, λ) = E(I, λ) +
∂Φ(q, I, λ)
∂t
=
= E(I, λ) +
˙
λ
∂Φ(q, I, λ)
∂λ
. ( 42.22)
q ∂Φ(q, I, λ)/∂λ
q = q(w, I)
Λ(I, w, λ) =
∂Φ(q, I, λ)
∂λ
q=q(w,I)
. (4 2 .2 3 )
Λ(I, w, λ)
Φ(q, I, λ) w
Φ(q, I, λ) λ
I
2πI
H
′
(I, w, λ) = E(I, λ) +
˙
λ Λ(I, w, λ) (42.24)
˙
I = −
˙
λ
∂Λ
∂w
, ˙w = ω +
˙
λ
∂Λ
∂I
. (42.25)
˙
λ
λ
Λ(I, w, λ)
w
H(p, x, ω) =
p
2
2m
+
1
2
mω
2
(t)x
2
. (42.26)
λ ω = ω(t)
Φ(q, I, λ)
ω
H
′
(I, w, ω) = ωI + ˙ω
∂Φ(q, I, ω)
∂ω
=
= ωI +
˙ω
2ω
p
2mωI − (mωx)
2
== ωI +
˙ω
2ω
I sin 2w , (42.27)
˙
I = −
˙ω(t)
ω(t)
I cos 2 w , ˙w = ω(t) +
˙ω(t)
2ω(t)
sin 2w . (42.28)
˙ω(t)
cos 2w
λ
“
∼
˙
λ T T
λ
¨
λ
˙
λ/T
˙
λ T ≪ λ ,
¨
λ T ≪
˙
λ . (43.1)
H(p, q, λ)
λ E
H(p, q, λ) = E . (43.2)
p = p(q, E, λ)
S(E, λ) =
I
p(q, E, λ) dq . (43.3)
λ E
hEi ≡
1
T
Z
T
0
E(t)dt (43.4)
λ(t)
E λ
E(t) λ(t)
I
S(E, λ) = 2πI(E, λ)
E
λ
I(E, λ) =
S(E, λ)
2π
=
1
2π
I
p(q, E, λ) dq . (43.5)
hI(E, λ)i = const (43.6 )
˙
λT
λ(t)
λ E
S
E
λ(t)
E
λ(t)
E
′
λ
′
˙ω ≪ ω
2
, ¨ω ≪ ˙ωω . (43.7)
H(p, x, ω) = E
√
2mE
p
2E/(mω
2
) S = 2πE/ω
I =
S
2π
=
E
ω
. (43.8)
l(t)
x = 0
0 l(t)
v
˙
l
x
x = l(t) T = 2l/v
v = |˙x|
˙
l ≪ v ,
¨
ll ≪
˙
lv . (43.9)
l 2mv = 2
√
2mE
I =
m
π
v l =
√
2m
π
√
E l . (43.10)