Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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F (q, Q, t)
δΣ =
t
2
Z
t
1
X
i
˙q
i
−
∂H
∂p
i
δp
i
−
˙p
i
+
∂H
∂q
i
δq
i
dt+
+
X
i
p
i
δq
i
|
t
2
t
1
= 0 (36.5)
δq
i
δp
i
˙q
i
=
∂H
∂p
i
, ˙p
i
= −
∂H
∂q
i
.
Σ
Σ =
t
2
Z
t
1
X
i
P
i
˙
Q
i
− H
′
(P, Q, t)
!
dt +
t
2
Z
t
1
dF (q, Q, t) ,
H
′
(P, Q, t) = H(p, q, t) +
∂F (q, Q, t)
∂t
, ( 36.6)
δΣ =
t
2
Z
t
1
X
i
˙
Q
i
−
∂H
′
∂P
i
δP
i
−
˙
P
i
+
∂H
′
∂Q
i
δQ
i
dt +
+
X
i
P
i
δQ
i
+
∂F
∂q
i
δq
i
+
∂F
∂Q
i
δQ
i
|
t
2
t
1
= 0 .
δQ
i
δP
i
˙
Q
i
=
∂H
′
∂P
i
,
˙
P
i
= −
∂H
′
∂Q
i
, (36.7)
H
′
(P, Q, t)
F (q, Q, t)
dF (q, Q, t) =
X
i
(p
i
dq
i
− P
i
dQ
i
) + (H
′
− H)dt . (36.8)
H(p, q, t)
F (q, Q, t)
F (q, Q, t) =
1
2
a q
2
+ 2b qQ + c Q
2
(36.9)
a, b c
p =
∂F
∂q
= a q + b Q , P = −
∂F
∂Q
= − b q − c Q .
Q = α q + β p , P = γ q + δ p , (36.10)
α = −
a
b
, β =
1
b
, γ =
ac − b
2
b
, δ = −
c
b
. (36.11)
∂(Q, P )
∂(q, p)
= αδ − βγ = 1 . (36.12)
H
′
(P, Q, t) = H(p, q, t) +
1
2
˙a q
2
+ 2
˙
b qQ + ˙c Q
2
, (36.13)
q p Q P
F (q, Q, t)
q, P Q, p
p, P
P
i
dQ
i
= d(P
i
Q
i
) − Q
i
dP
i
,
Φ(q, P, t) = F +
s
X
i=1
Q
i
P
i
, (36.14)
dΦ(q, P, t) =
s
X
i=1
(p
i
dq
i
+ Q
i
dP
i
) + (H
′
− H) dt . (36.15)
Φ(q, P, t)
p
i
=
∂Φ
∂q
i
, Q
i
=
∂Φ
∂P
i
, H
′
= H +
∂Φ
∂t
. (36.16)
F (q, Q, t) =
s
X
i=1
q
i
Q
i
Q
i
= p
i
, P
i
= − q
i
.
“ “ p, q
Φ(q, P, t) =
s
X
i=1
q
i
P
i
Q
i
= q
i
, P
i
= p
i
.
Φ(q, P, t) =
P
i
q
i
P
i
+ εW ( q, P ) ε
Φ(q, P, t) = qP + δτH(P, q, t) , (36.17)
ε ≡ δτ H(p, q, t)
p =
∂Φ
∂q
= P + δτ
∂H(P, q, t)
∂q
,
Q =
∂Φ
∂P
= q + δτ
∂H(P, q, t)
∂P
.
δτ
P (t) = p −δτ
∂H(p, q, t)
∂q
= p(t) + ˙pδτ = p(t + δτ ) ,
Q(t) = q + δτ
∂H(p, q, t)
∂p
= q( t) + ˙qδτ = q(t + δτ) .
Q(t) = q(t + δτ) , P (t) = p(t + δτ) (36.18)
δτ
H
′
(P, Q, t) = H(p, q, t) +
∂Φ
∂t
= H(p, q, t) + δτ
∂H(P, q, t)
∂t
≈
≈ H(p, q, t) + δτ
dH(p, q, t)
dt
= H(p, q, t + δτ)
δτ
Q(t) = q(t + τ ) , P (t) = p(t + τ) (36.19)
Φ(r, P) = rP + δaP
Φ(r, P) = rP + δϕ[r, P]
r δa δϕ
Φ(x, y, P
x
, P
y
) = xP
x
+ yP
y
+ ε(xy + P
x
P
y
) ,
ε → 0
f h
q, p t
t = const
h(q, p, t) “
q
i
p
i
“ τ
t “
q
i
p
i
τ
f(q(τ ), p(τ), t) τ
df
dτ
= {h, f}
p,q
≡
X
i
∂h
∂p
i
∂f
∂q
i
−
∂h
∂q
i
∂f
∂p
i
. (37.1)
q, p → Q, P
“ τ
“
h
′
(Q, P, t) = h(q(Q, P, t), p(Q, P, t), t) , (37.2)
τ f(q(Q, P, t), p(Q, P, t), t)
df
dτ
= {h
′
, f}
P,Q
≡
X
i
∂h
′
∂P
i
∂f
∂Q
i
−
∂h
′
∂Q
i
∂f
∂P
i
. (37.3)
{h, f}
p,q
= {h, f}
P,Q
. (3 7 .4 )
q, p → Q, P
{Q
i
, Q
j
}
p,q
= 0, {P
i
, P
j
}
p,q
= 0, {P
i
, Q
j
}
p,q
= δ
ij
. (3 7 .5 )
{Q
i
, Q
j
}
p,q
= {Q
i
, Q
j
}
P,Q
=
X
k
∂Q
i
∂P
k
∂Q
j
∂Q
k
−
∂Q
i
∂Q
k
∂Q
j
∂P
k
= 0 ,
(37.6)
∂Q
l
/∂P
k
= 0
q, p
Q, P
X
i
(p
i
dq
i
− P
i
dQ
i
)
t = const F
Q, P
X
i
(p
i
dq
i
−P
i
dQ
i
) =
X
ik
p
i
∂q
i
∂Q
k
dQ
k
+
∂q
i
∂P
k
dP
k
−
X
i
P
i
dQ
i
.
(37.7)
dF
∂
2
F
∂Q
j
∂Q
l
=
∂
2
F
∂Q
l
∂Q
j
,
∂
2
F
∂P
j
∂P
l
=
∂
2
F
∂P
l
∂P
j
, (37 .8 )
∂
2
F
∂P
j
∂Q
l
=
∂
2
F
∂P
l
∂Q
j
.
∂
∂P
j
X
i
p
i
∂q
i
∂Q
l
− P
l
!
=
∂
∂Q
l
X
i
p
i
∂q
i
∂P
j
!
[P
j
, Q
l
]
p,q
≡
X
i
∂p
i
∂P
j
∂q
i
∂Q
l
−
∂p
i
∂Q
l
∂q
i
∂P
j
= δ
jl
. (37.9a)
[f, g]
p,q
[Q
j
, Q
l
]
p,q
= 0, [P
j
, P
l
]
p,q
= 0 . (37.9b)
Q, P
“ u
1
= Q, u
2
=
Q
2
, . . . , u
s
= Q
s
, u
s+1
= P
1
, . . . , u
2s
= P
s
2s × 2s
ˆ
L
ˆ
P
L
ij
= [u
i
, u
j
], P
ij
= {u
i
, u
j
}; i, j = 1, 2, . . . , 2s .
ˆ
L
ˆ
P =
ˆ
E
ˆ
L
−1
=
ˆ
P
ˆ
E
Q = α x + β p, P = γ x + δ p , (38.1)
α, β, γ, δ
{P, Q}
p,x
= αδ − βγ = 1 .