Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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1 2 3 4
z
xz
z = 0
xz
θ
x
= p
x
/p
z
|θ
x
| ≪ 1
p
z
≈ const
θ
x
H(p
x
, p
z
, x, z, t)
p
z
= const z = p
z
t/m
x, p
x
x
1
2
x
p
x
3 4
x
p
x
x, p
x
z
x
x
0
< x <
x
0
+ ∆x
0
p
0
< p < p
0
+ ∆p
0
x
S(q, t, q
(1)
, t
1
)
∂S
∂t
= − H(p
1
, p
2
, . . . , p
s
, q
1
, q
2
, . . . , q
s
, t)
p
i
p
i
= ∂S/∂q
i
∂S
∂t
+ H
∂S
∂q
1
, . . . ,
∂S
∂q
s
, q
1
, . . . , q
s
, t
= 0 , (41.1)
s
α
1
, . . . , α
s
s + 1
α
s+1
S = f(q
1
, . . . , q
s
, α
1
, . . . , α
s
, t) + α
s+1
. (41.2)
q
i
(t) p
i
(t)
f(q, α, t)
α
i
q
i
= q
i
(α, β, t) , p
i
= p
i
(α, β, t) (41.3)
∂f
∂q
i
= p
i
,
∂f
∂α
i
= β
i
, (41.4)
β
i
H
′
(α, β, t) = H +
∂f
∂t
H
′
(α, β, t) = 0
˙α
i
= −
∂H
′
∂β
i
= 0 ,
˙
β
i
= −
∂H
′
∂α
i
= 0
α, β
S
S(q
1
, . . . , q
s
, t) = − Et + S
0
(q
1
, . . . , q
s
) , (41.5)
S
0
(q
1
, . . . , q
s
)
H
∂S
0
∂q
1
, . . . ,
∂S
0
∂q
s
, q
1
, . . . , q
s
= E . (41.6)
S
0
S
0
(q
1
, . . . , q
s
) =
s
X
i=1
S
i
(q
i
) , (41.7a)
S
i
(q
i
) =
Z
p
i
(q
i
) dq
i
. (41.7b)
U(r) = −
α
r
H(p, r, t) =
p
p
2
c
2
+ m
2
c
4
−
α
r
,
p =
mv
p
1 − (v/c)
2
.
E M
E =
p
p
2
c
2
+ m
2
c
4
−
α
r
, M = [r, p] =
m [r, v]
p
1 − (v/c)
2
.
M
r ϕ
L = −mc
2
s
1 −
˙r
2
+ r
2
˙ϕ
2
c
2
+
α
r
.
p
r
=
∂L
∂ ˙r
=
m ˙r
p
1 − (v/c)
2
, p
ϕ
≡ M =
∂L
∂ ˙ϕ
=
mr
2
˙ϕ
p
1 − (v/c)
2
p
2
= p
2
r
+
M
2
r
2
,
p
r
(r) = ±
s
1
c
2
E +
α
r
2
−
M
2
r
2
− m
2
c
2
.
S
0
(r, ϕ) =
Z
p
ϕ
(ϕ) dϕ +
Z
p
r
(r) dr .
S(r, ϕ, E, M, t) = −E t + Mϕ +
Z
p
r
(r) dr ,
E M α
1,2
∂S/∂α
1
= β
1
∂S/∂E = const
r(t)
t =
1
c
2
Z
E +
α
r
dr
p
r
(r)
, (41.8)
∂S/∂M = const
ϕ =
Z
M
r
2
dr
p
r
(r)
(41.9)
U(r) = −
α
r
+
β
r
2
.
m →
E
c
2
, E →
E
2
− m
2
c
4
2E
, β → −
α
2
2E
. (41.10)
U(r) = −α/r
δU(r) = −
α
2
2E r
2
. (41.11)
M > α/c
r =
˜p
1 + ˜e cos(γϕ)
, (41.12)
γ =
s
1 −
α
2
c
2
M
2
, ˜p =
c
2
M
2
− α
2
Eα
,
˜e =
r
1 +
(E
2
− m
2
c
4
)(c
2
M
2
− α
2
)
E
2
α
2
. (41.13)
E < mc
2
˜e < 1
∆ϕ =
2π
γ
, (41.14)
δϕ =
2π
γ
− 2π . (41.15a)
v ≪ c mc
2
− E ≈ α/(2a) ≪ mc
2
δϕ ≈ π
α/a
mc
2
(1 − e
2
)
, (41.15b)
a e
δϕ ≈ 7, 2
′′
(41.16)
δϕ = (43, 11 ± 0, 45)
′′
(41.17)
δϕ
δϕ
OTO
= 6δϕ ≈ 43
′′
. (41.18)
n(r)
Ψ(r, t) = const
Ψ(r, t)
(∇Ψ)
2
=
n
2
c
2
∂Ψ
∂t
2
,
c
33.12a
H(p, r) =
c
n(r)
|p|.
∂S
∂t
+
c
n(r)
|∇S| = 0
S(r, t) Ψ(r, t)
S(r, t) = const S(r , t)
U(r, θ, ϕ) =
a cos θ
r
2
Eρ
2
≪ a
z
w
1
, . . . , w
s
I
1
, . . . , I
s
∂H/∂t = 0
H(p, q)
w I
q p
I
w
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