Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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N
F
α
(r
1
, ..., r
N
, t) = 0, α = 1, ..., n, (B.1)
F
α
n
F
1
(r, t) ≡ r − l(t) = 0.
F
a
= −
∂U
∂r
a
R
a
δr
a
N
X
a=1
R
a
· δr
a
= 0 . (B.2)
t
m
a
¨
r
a
= −
∂U
∂r
a
+ R
a
, a = 1, . . . , N , (B.3)
r
a
(t) R
a
(t)
s = 3N − n
N
3N K s =
3N − n K
0
K
K
0
n
˜q
α
= F
α
(r
1
, . . . , r
N
, t), α = 1, . . . , n ,
q
i
i = 1, . . . , s
K
0
n
˜q
α
= 0, α = 1 , . . . , n, (B.4)
q
i
K
0
˜
U(˜q
1
, . . . , ˜q
n
) =
X
α
˜
U
α
(˜q
α
) , (B.5 )
˜
U
α
(˜q
α
)
(−∂
˜
U/∂ ˜q
α
)
K
0
˜
L = L
1
(q, ˜q, ˙q,
˙
˜q, t) −
˜
U(˜q), (B.6)
L
1
˜q
α
R
α
=
−∂
˜
U/∂ ˜q
α
˜q
α
R
α
q
i
˜
U ˜q
α
= 0
˙
˜q
α
= 0
L(q, ˙q, t) = L
1
(q, 0, ˙q, 0, t) = T (q, ˙q, t) − U(q, t) . (B.7)
d
dt
∂L(q, ˙q, t)
∂ ˙q
i
=
∂L(q, ˙q, t)
∂q
i
, i = 1, 2, ..., s = 3N − n.
R
a
= m
a
¨
r
a
+
∂U
∂r
a
N
X
a=1
m
¨
r
a
+
∂U
∂r
a
· δr
a
= 0 .
r
a
δr
a
q
i
δq
i
δq
i
a
x
a
x
ϕ
.
.
˙x + a ˙ϕ = 0,
x ϕ
x + aϕ = const ,
V + [Ω , a] = 0 (B.8)
a
A
C
B
D
˜q
A, B, C, D
BD = const
AC = const
BD
AC
AC
BD
s + n
L = L(q
1
, . . . , ˙q
s+n
, t)
n
F
α
(q
1
, . . . , q
s+n
, t) = 0, α = 1, ..., n . (B.9)
s+n
X
i=1
c
αi
˙q
i
+ c
α
= 0, (B.10)
c
αi
=
∂F
α
∂q
i
, c
α
=
∂F
α
∂t
. (B.11)
n
λ
α
(t)
˜
L(q
1
, . . . , ˙q
s+n
, t) =
= L(q
1
, . . . , ˙q
s+n
, t) +
n
X
α=1
λ
α
(t) F
α
(q
1
, . . . , q
s+n
, t) , (B.12)
˜
S =
R
˜
Ldt
s + n
q
i
(t)
s + n n
q
i
(t) s+2n
d
dt
∂L
∂ ˙q
i
−
∂L
∂q
i
=
l
X
α=1
λ
α
c
αi
. (B.13)
δq
i
s+n
X
i=1
c
αi
δq
i
= 0 . (B.14)
δq
i
∂c
αi
∂q
j
=
∂c
αj
∂q
i
,
∂c
αi
∂t
=
∂c
α
∂q
i
, (B.15)
α, i, j
α δA
12
(q
1
, q
2
) → (q
1
+ δq
1
, q
2
) → (q
1
+ δq
1
, q
2
+ δq
2
)
δA
21
(q
1
, q
2
) → (q
1
, q
2
+ δq
2
) → (q
1
+ δq
1
, q
2
+ δq
2
)
δq
1
δq
2
δA
12
= λ
α
[c
α1
(q
1
, q
2
, . . .)δq
1
+ c
α2
(q
1
+ δq
1
, q
2
, . . .)δq
2
] =
= λ
α
[c
α1
(q
1
, q
2
, . . .)δq
1
+c
α2
(q
1
, q
2
, . . .)δq
2
+
∂c
α2
(q
1
, q
2
, . . .)
∂q
1
δq
1
δq
2
] ,
δA
21
1 ⇄ 2
δA
12
− δA
21
= λ
α
∂c
α2
∂q
1
−
∂c
α1
∂q
2
δq
1
δq
2
.
δA
12
− δA
21
= 0
δq
n
n
¨x + ω
2
(t) x = 0 , (C.1)
ω(t)
ω(t + T ) = ω(t). (C.2)
x
1
(t) x
2
(t)
W (x
1
, x
2
) =
x
1
x
2
˙x
1
˙x
2
= x
1
(t) ˙x
2
(t) − x
2
(t) ˙x
1
(t).