Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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x ˙x =
˙q U(q
0
)
L(x, ˙x) =
1
2
m ˙x
2
−
1
2
kx
2
. (19.4)
m¨x + kx = 0 (19.5)
x = A cos(ωt + ϕ) (1 9 .6 )
−mω
2
+ k = 0 , (19.7)
ω =
r
k
m
. (19.8)
A ωt + ϕ ϕ
ω
T =
2π
ω
. (19.9)
m k ω
2
=
k/m
x A
L = T − U(q
1
, q
2
, . . . , q
s
) , T =
1
2
s
X
ij=1
a
ij
(q) ˙q
i
˙q
j
≥ 0 . (19.10)
˙q
i
˙q
j
i ↔ j a
ij
(q)
a
ij
(q) = a
ji
(q) . (19.11)
q
i0
(i = 1, 2, . . . , s)
x
i
= q
i
−q
i0
U(q) =
1
2
X
ij
k
ij
x
i
x
j
+ const , k
ij
=
∂
2
U
∂q
i
∂q
j
|
q
l0
= k
ji
. (19.12)
x
i
= 0
X
ij
k
ij
x
i
x
j
≥ 0 . (19.13)
a
ij
(q) q
i0
a
ij
(q) = m
ij
+ O(x
k
); m
ij
= a
ij
(q
k0
) = m
ji
. (1 9 .1 4)
X
ij
m
ij
˙x
i
˙x
j
≥ 0 (19.15)
x
i
˙x
i
= ˙q
i
L =
1
2
X
ij
(m
ij
˙x
i
˙x
j
− k
ij
x
i
x
j
) . (19.16 )
x =
x
1
x
2
x
s
,
ˆm =
m
11
m
12
. . . m
1s
m
21
m
22
. . . m
2s
. . . . . . . . . . . .
m
s1
m
s2
. . . m
ss
,
ˆ
k =
k
11
k
12
. . . k
1s
k
21
k
22
. . . k
2s
. . . . . . . . . . . .
k
s1
k
s2
. . . k
ss
(x, y) =
s
X
i=1
x
i
y
i
.
L(x,
˙
x) =
1
2
(
˙
x, ˆm
˙
x) −
1
2
(x,
ˆ
k x) , (19.17)
(
˙
x, ˆm
˙
x) ≥ 0 , (x,
ˆ
k x) ≥ 0, ˆm
T
= ˆm,
ˆ
k
T
=
ˆ
k , (19.18)
ˆm
T
ˆm
d
dt
∂L
∂
˙
x
=
∂L
∂x
ˆm
¨
x +
ˆ
kx = 0 . (19.19)
x = A cos(ωt + ϕ) (19.20)
−ω
2
ˆm +
ˆ
k
A = 0 . (19.21)
−ω
2
ˆm +
ˆ
k
= 0 . (19.22)
ω
2
1
, ω
2
2
, . . . , ω
2
s
ω
α
ω
2
α
A
(α)
−ω
2
α
ˆm +
ˆ
k
A
(α)
= 0 . (19.23)
A
(α)
aA
(α)
a
ω
2
α
ω
α
x
(α)
(t) = A
(α)
Q
α
(t) , Q
α
(t) = a
α
cos(ω
α
t + ϕ
α
) , (19.24)
x =
s
X
α=1
A
(α)
Q
α
(t) , (19.25)
x
i
=
s
X
α=1
A
(α)
i
Q
α
(t) . (19.26)
s a
α
ϕ
α
x(0)
˙
x(0)
|ϕ
i
| ≪ 1 l
1
= l
2
= l
L =
1
2
ml
2
(8 ˙ϕ
2
1
+ 4 ˙ϕ
1
˙ϕ
2
+ ˙ϕ
2
2
) −
1
2
mgl(4ϕ
2
1
+ ϕ
2
2
) .
ˆm = ml
2
8 2
2 1
,
ˆ
k = mgl
4 0
0 1
.
8 ¨ϕ
1
+ 2 ¨ϕ
2
+ 4ω
2
0
ϕ
1
= 0, 2 ¨ϕ
1
+ ¨ϕ
2
+ ω
2
0
ϕ
2
= 0, ω
0
=
p
g/l
x =
ϕ
1
ϕ
2
=
A
1
A
2
cos(ωt + χ).
A
i
(−8ω
2
+ 4ω
2
0
)A
1
− 2ω
2
A
2
= 0 , −2ω
2
A
1
+ (−ω
2
+ ω
2
0
)A
2
= 0 .
ω
4
− 3ω
2
ω
2
0
+ ω
4
0
= 0 ,
ω
1,2
=
s
3 ∓
√
5
2
ω
0
x
(1)
= A
(1)
Q
1
(t) , x
(2)
= A
(2)
Q
2
(t) ; Q
α
(t) = a
α
cos(ω
α
t+χ
α
) .
ϕ
1
ϕ
2
A
(1)
=
1
√
5 − 1
, A
(2)
=
−1
√
5 + 1
Q
1
Q
2
ϕ
1
ϕ
2
A
(1)
A
(2)
ω
1
A
(1)
ω
2
A
(2)
ϕ
1
= Q
1
− Q
2
,
ϕ
2
= (
√
5 − 1)Q
1
+ (
√
5 − 1) Q
2
A
(1)
A
(2)
Q
1
Q
2
◦
A
(1)
, A
(2)
= 3 6= 0 .
A
(1)
,
ˆ
kA
(2)
=
A
(1)
, ˆmA
(2)
= 0 . (19.27)
A
(1)
,
ˆ
k
mgl
A
(2)
!
= (1 ,
√
5 − 1)
4 0
0 1
−1
√
5 + 1
=
= (1,
√
5 − 1)
−4
√
5 + 1
= 0 .
ω
α
ω
β
ω
α
6= ω
β
A
(α)
A
(β)
ω
2
α
ˆm A
(α)
=
ˆ
k A
(α)
, ω
2
β
ˆm A
(β)
=
ˆ
k A
(β)
. (20.1)
A
(α)
ω
2
β
A
(α)
, ˆm A
(β)
=
A
(α)
,
ˆ
k A
(β)
. (20.2)
A
(β)
ω
2
α
A
(β)
, ˆm A
(α)
− ω
2
β
A
(α)
, ˆm A
(β)
=
=
A
(β)
,
ˆ
k A
(α)
−
A
(α)
,
ˆ
k A
(β)
. (20.3)
ˆn
A, ˆn B) = (ˆn
T
A, B
ˆ
k ˆm
ω
2
α
− ω
2
β
A
(α)
, ˆm A
(β)
= 0 . (20.4)
ω
2
α
− ω
2
β
6= 0
A
(α)
, ˆm A
(β)
= 0 (20.5 a)
A
(α)
,
ˆ
k A
(β)
= 0 . (20.5b)
x
(α)
=
A
(α)
Q
α
x
(β)
= A
(β)
Q
β
m
ij
k
ij
x
(α)
x
(β)
“ “
“ “
x
(1)
x
(2)
ω
1
= ω
2
c
1
x
(1)
+ c
2
x
(2)
c
1
c
2
x
(1)
x
(2)
Q
α
x = (x
1
, . . . , x
i
, . . . , x
s
)
Q = (Q
1
, . . . , Q
α
, . . . , Q
s
)
x =
ˆ
UQ , x
i
=
X
α
U
iα
Q
α
, U
iα
≡ A
(α)
i
,
L =
s
X
α=1
L
α
, L
α
=
1
2
M
α
˙
Q
α
−
1
2
K
α
Q
2
α
,
M
α
=
A
(α)
, ˆm A
(α)
, K
α
=
A
(α)
,
ˆ
k A
(α)
, (20.6)
M
α
¨
Q
α
+ K
α
Q
α
= 0 ( 20.7)
Q
α
ω
α
x
i
ω
2
α
=
K
α
M
α
=
A
(α)
,
ˆ
k A
(α)
A
(α)
, ˆm A
(α)
≥ 0 , (20.8)
ω
α
A
(α)
m
1
= m
2
= m
l
1
l
2
k
ϕ
1
ϕ
2
|ϕ
1,2
| ≪ 1
x = l
1
ϕ
1
, y = l
2
ϕ
2