Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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ϕ x, p/(mω)
x = A cos(ωt + ϕ
0
) , p = −mωA sin(ωt + ϕ
0
) (38.2)
ϕ = ωt
Q
x
P/(mω)
p/(mω)
ϕ
ϕ
x, p/(mω)
Q = A cos ϕ
0
, P = −mωA sin ϕ
0
,
H
′
= 0.
a =
mωx + ip
√
2mω
, a
∗
=
mωx − ip
√
2mω
(38.3)
Q = a, P = ia
∗
(38.4)
Q = a =
r
mω
2
A e
−i(ωt+ϕ
0
)
, P = ia
∗
= i
r
mω
2
A e
+i(ωt+ϕ
0
)
,
(38.5a)
H
′
= ωa
∗
a = − iωQP. (38.5b)
ae
iωt
ia
∗
e
−iωt
.
H
′
= 0
z
P
X
= p
x
= mv
x
, X = x −
p
y
mω
= −
v
y
ω
,
P
Y
= p
y
= mωx
0
, Y = y −
p
x
mω
= y
0
x, y, p
x
, p
y
H(P
X
, P
Y
, X, Y ) =
mv
2
2
=
P
2
X
2m
+
mω
2
X
2
2
.
x, p
y
/(mω) y, p
x
/(mω)
H =
p
2
2m
+
mω
2
x
2
2
+ αx
3
|αx| ≪ mω
2
Φ(x, P, t) = xP + ax
3
+ bx
2
P + cxP
2
+ dP
3
.
a = c = 0
b d
Q P x(t)
t
1
q
(1)
a, b, c . . .
q t
a
S
12
=
2
Z
1
L(q, ˙q, t) dt , (39.1a)
a
t
1
q
(1)
S
12
= S(q, t) . (39.1b)
q(t)
t
1
t t + dt
q
(1)
q
q + dq
1
2
4
3
b
a
c
S(q, t)
S(q, t)/~
~
S(q, t)
S( q, t)
S(q, t)
∂S/∂t
q t + dt
b
S(q, t + dt) 1 → 2
a
S(q, t) 2 → 3
Ldt
L = p ˙q − H
2 → 3 ˙q = 0
Ldt|
2→3
= − Hdt .
δS = 0
∼ dt
S(q, t + dt) = S(q, t) − Hdt,
S(q, t + dt) − S(q, t) = −Hdt,
∂S
∂t
= −H . (39.2)
∂S/∂q S(q, t)
(q, t) (q +dq, t)
S(q + dq, t) =
4
Z
1
Ldt
c “
1 → 2 a 2 → 4
1 → 2 → 4 “
2 → 4
˙q
q(t)
2 → 4
dt → 0 , ˙q → ∞, ˙qdt = dq .
Ldt|
2→4
= (p ˙q − H)dt = pdq − Hdt
pdq
S(q + dq, t) = S(q, t) + pdq,
∂S/∂q = p
∂S
∂q
i
= p
i
. (39.3)
q
(1)
i
t
1
dS(q, t, q
(1)
, t
1
) =
X
i
p
i
dq
i
− Hdt −
X
i
p
(1)
i
dq
(1)
i
+ H
(1)
dt
1
.
(39.4)
S(q, t, q
(1)
, t
1
)
q
i
(t)
τ
F (q
(1)
, q, t) = −S(q, t + τ, q
(1)
, t) , (39.5)
q
(1)
q
H
(1)
H
S(q, t, q
(1)
, t
1
)
q
i
= g
i
(t) , i = 1, . . . , s
S
12
≡ S(q
(2)
, t
2
, q
(1)
, t
1
) =
Z
2
1
L
g(t) ,
dg(t)
dt
, t
dt .
q
′
i
t
′
q
′
i
= g
i
(t
′
)
q
i
t
ε
q
i
+ δq
i
= g
i
(t + δt) ,
δq
i
= εf
i
(q(t), t) , δt = εh(q(t), t) .
S
1
′
2
′
=
Z
2
′
1
′
L
g(t
′
),
dg(t
′
)
dt
′
, t
′
dt
′
=
= S(q
(2)
+ δq
(2)
, t
2
+ δt
2
, q
(1)
+ δq
(1)
, t
1
+ δt
1
) .
g(t) 12
g(t
′
) 1
′
2
′
S
1
′
2
′
− S
12
=
=
X
i
p
i
(t
2
)δq
(2)
i
−E( t
2
)δt
2
−
X
i
p
i
(t
1
)δq
(1)
i
+ E(t
1
)δt
1
. (39.6)
E(t) q
i
(t)
p
i
(t)
S
12
= S
1
′
2
′
E(t) · h(q(t), t) −
s
X
i=1
p
i
(t) · f
i
(q(t), t) = const . (39.7)
2s − 1
2s
p
1
, . . . p
s
, q
1
, . . . q
s
Ω
Γ =
Z
Ω
dp
1
. . . dp
s
dq
1
. . . dq
s
.
q, p
Q, P Ω Ω
′
Γ =
Z
Ω
′
D dP
1
. . . dP
s
dQ
1
. . . dQ
s
D =
∂(p, q)
∂(P, Q)
.
p
p
1
, . . . , p
s
q, P Q Ω
′
Γ
′
=
Z
Ω
′
dP
1
. . . dP
s
dQ
1
. . . dQ
s
.
D = 1
Γ
′
= Γ
p, q P, q
P, Q
D =
∂(p, q)
∂(P, q)
∂(P, q)
∂(P, Q)
.
D =
∂(p)
∂(Q)
q=const
.
∂(q)
∂(Q)
P =const
. (40.1)
Φ(q, P )
∂p
i
∂P
k
=
∂Φ
∂q
i
∂P
k
,
∂Q
i
∂q
k
=
∂Φ
∂q
k
∂P
i
.
D = 1
z
z
x p
x