Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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dx ∧ dy =
∂x
∂ξ
dξ +
∂x
∂η
dη
∧
∂y
∂ξ
dξ +
∂y
∂η
dη
=
=
∂x
∂ξ
∂y
∂ξ
dξ ∧dξ +
∂x
∂η
∂y
∂ξ
dη ∧dξ +
∂x
∂ξ
∂y
∂η
dξ ∧dη +
∂x
∂η
∂y
∂η
dη ∧dη =
=
∂x
∂ξ
∂y
∂η
−
∂x
∂η
∂y
∂ξ
dξ ∧ dη = Ddξ ∧ dη ,
dxdy Ddξdη.
n
dx
1
∧ dx
2
∧ ... ∧ dx
n
x
i
= x
i
(ξ
1
, ξ
2
, . . . , ξ
n
), i = 1, 2, . . . , n ,
dx
i
=
X
j
∂x
i
∂ξ
j
dξ
j
, (D.4)
dx
1
∧ dx
2
∧ . . . ∧ dx
n
=
∂(x
1
, x
2
, . . . , x
n
)
∂(ξ
1
, ξ
2
, . . . , ξ
n
)
dξ
1
∧ dξ
2
∧ . . . ∧ dξ
n
.
(D.5)
n
∂x
i
/∂ξ
j
n i
n
j
dξ
j1
∧ dξ
j2
∧ . . . ∧ dξ
jn
.
j
j1, j2, . . . , jn
1, 2, . . . , n
∂x
i
/∂ξ
j
X
i,j,...,k
F
i,j,...,k
(x
1
, x
2
, . . . , x
n
) , dx
i
∧ dx
j
∧ . . . dx
k
l
l
ω
l
Z
A dS
ω
2
= A
x
dy ∧ dz + A
y
dz ∧ dx + A
z
dx ∧ dy.
l
l + 1
dω
l
= ω
l+1
=
X
i,j,...,k
dF
i,j,...,k
∧ dx
i
∧ dx
j
∧ . . . dx
k
,
dF
i,j,...,k
dF
i,j,...,k
=
n
X
m=1
∂F
i,j,...,k
∂x
m
dx
m
. (D.6)
∂
2
F
∂x
i
∂x
j
=
∂
2
F
∂x
j
∂x
i
.
dω
l
= 0 ω
l
ω
l
= dω
l−1
ω
1
= f( x) dx
dx
ω
1
x
dx
dx
1
, dx
2
, ..., dx
s
∧
s
X
i=1
p
i
dq
i
−
s
X
i=1
P
i
dQ
i
= dF (D.7)
(dF )
s
X
i=1
dp
i
∧ dq
i
=
s
X
i=1
dP
i
∧ dQ
i
. (D.8)
F (q
1
, ..., Q
s
)
s
s!
s
Y
i=1
dp
i
∧ dq
i
= s!
s
Y
i=1
dP
i
∧ dQ
i
.
∂(Q
1
, P
1
, ..., Q
s
, P
s
)
∂(q
1
, p
1
, ..., q
s
, p
s
)
= 1 . ( D.9)
f g
f = f(q, p), g = g(q, p) .
df ∧ dg ∧ (ω)
s−1
, (D.10)
ω
(s − 1)! {f, g}
s
Y
i=1
dp
i
∧ dq
i
, (D.11)
{f, g}
p, q
P, Q
p, q P, Q
{f, g}
P,Q
(ω)
s−1
ω s − 1
