Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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f(X
n
+ l) = λf(X
n
) = λ
2
f(X
n−1
) = . . . = λ
n
f(X
1
) . (26.5b)
a
ω
2
= ω
2
0
2 − λ −
1
λ
, (26.6)
ω λ
λ
1,2
= d ±
p
d
2
− 1 , d = 1 −
ω
2
2ω
2
0
.
λ
1
λ
2
= 1 ω < 2ω
0
d < 1
λ
1,2
λ
1
= λ
∗
2
|λ
1,2
| = 1
ω > 2ω
0
d
2
> 1 λ
1,2
λ
1,2
< 0 |λ
1
| > 1 |λ
2
| < 1
|λ| = 1 λ
λ = e
∓iϕ
, (26.7a)
ω
2
= 4ω
2
0
sin
2
ϕ
2
(26.8)
y
n
= R e
h
Ae
i(ωt∓nϕ)
i
. (26.9a)
K = ϕ/l
λ = e
∓iKl
(26.7b)
X X
y
n
= R e
h
Ae
i(ωt∓KX
n
)
i
. (26.9b)
ω T =
2π/ω K “
Λ =
2π
K
=
2πl
ϕ
. (26.10)
ϕ
0 < ω < 2ω
0
ωt ∓ KX
n
= const
x
X
n
= ±
ω
K
t −
const
K
. (26.11)
λ < 0 λ
λ = −e
±ψ
= − e
∓κl
, ψ = κl , (26.12)
ω
2
= 4ω
2
0
ch
2
ψ
2
, (26.13)
X
n
y
n
= R e
(−1)
n
A e
iωt∓κX
n
= R e
(−1)
n
A e
∓nψ
e
iωt
. (26.14)
ω > 2ω
0
ϕ → π − iψ. (26.15)
(n − 1)
n dt F
n
=
−mω
2
0
(y
n
−y
n−1
) n
n dy
n
= ˙y
n
dt
dE = F
n
dy
n
= − mω
2
0
(y
n
− y
n−1
) ˙y
n
dt.
X
dE
dt
= −mω
2
0
(y
n
− y
n−1
) ˙y
n
.
dE
dt
= ±
1
2
mω
2
0
ω|A|
2
sin ϕ .
y
n
= A
(+)
e
i(ωt+nϕ)
+ A
(−)
e
i(ωt−nϕ)
.
y
0
= 0 A
(−)
= − A
(+)
y
n
= R e
2iA
(+)
sin nϕ e
iωt
= A sin nϕ cos(ωt + χ) ,
2iA
(+)
= A e
iχ
, (26.16)
n
y
N+ 1
= 0
sin(N + 1)ϕ = 0
sin(N + 1)ϕ = 0 N
ϕ
α
=
πα
N + 1
, α = 1, 2 , ..., N , (26.17)
α = 0 α = N + 1 y
n
= 0
α = N + s ϕ
N+ s
= 2π − ϕ
N
s
+ 2
α = N + s α = N − s + 2
N
ω
α
= 2ω
0
sin
πα
2(N + 1 )
, α = 1, 2 , . . . , N . (26.18)
N
1 2 N
α
ω
α
2ω
0
0 < ω
α
<
2ω
0
ω > 2ω
0
α
y
(α)
= const ·
sin ϕ
α
sin 2ϕ
α
sin Nϕ
α
Q
α
(t); Q
α
(t) = a
α
cos(ω
α
t + χ
α
) .
(26.19)
ϕ
α
Λ
α
=
2π
K
α
=
2πl
ϕ
α
=
2L
α
, α = 1, 2 , ..., N , (26.20)
L = l · (N + 1) α
α
const =
1
s
N
P
n=1
sin
2
nϕ
α
=
r
2
N + 1
,
(y
(α)
, y
(β)
) = δ
αβ
Q
2
α
.
y
n
=
N
X
α=1
U
nα
Q
α
(t) , U
nα
=
r
2
N + 1
sin
πnα
N + 1
, (26.21)
Q
α
L =
N
X
α=1
L
α
; L
α
=
m
2
˙
Q
2
α
− ω
2
α
Q
2
α
, (26.22)
N
N
m
k
AB x
n
n
x
0
= x
N+ 1
≡ 0
¨x
n
+ ω
2
0
(2x
n
− x
n−1
− x
n+1
) = 0 , n = 1, 2, . . . , N , (2 6 .2 3)
ω
0
=
p
k/m
y
n
→ x
n
N
m
k
k
m
x
1
x
2
x
N
x
N
A
m M 2N
k
AB A B
x
n
n
k k
m M m M
x
1
x
2
x
2N
A B
m M
m¨x
2n−1
+ k(2x
2n−1
− x
2n−2
− x
2n
) = 0 ,
M ¨x
2n
+ k(2x
2n
− x
2n−1
− x
2n+1
) = 0,
x
0
= x
2N+1
≡ 0 n = 1, 2, . . . N
x
2n−1
= A
α
sin [(2n − 1)ϕ
α
] cos(ω
α
t + χ
α
) ,
x
2n
= B
α
sin [2nϕ
α
] cos(ω
α
t + χ
α
) ,
B
α
=
2k −mω
2
α
2k cos ϕ
α
A
α
, α = 1 , 2, . . . , N .
ω
2
(∓)α
=
k
µ
1 ∓
r
1 −
4µ
2
mM
sin
2
ϕ
α
!
,
µ =
mM
m + M
, ϕ
α
=
πα
2N + 1
, α = 1, 2, . . . N .
ω
(−)α
0 < ω
(−)α
<
r
2k
M
, (27.1)
ω
(+)α
r
2k
m
< ω
(+)α
<
s
2k
µ
, (27.2)
1 2 N
α
ω
(+)
ω
(−)
q
2k
µ
q
2k
m
q
2k
M
ω
r
2k
M
< ω <
r
2k
m
(27.3)
m ≪ M
ω >
s
2k
µ
A
(−)α
B
(−)α
(2n − 1) (2n) m M
A
(+)α
B
(+)α
n
n
x
n
x
n
а)
б)
N = 10 α = 2 M = 2m
N = 10 α = 2 M = 2m
A
(−)α
B
(−)α
B
y
B
= b cos(γt + χ)
y
y
0
≡ 0 , y
N+ 1
= b cos(γt + χ) . (28.1)
γ < 2ω
0
y
n
= B sin nϕ cos(γt + χ) , (28.2)
y
0
= 0
ϕ γ
γ
2
= 4ω
2
0
sin
2
ϕ
2
, (2 8 .3 )
B sin(N + 1)ϕ = b ,
y
n
= b
sin nϕ
sin(N + 1)ϕ
cos(γt + χ) , (28.4)
γ ≪ ω
0
ϕ ≈ γ/ω
0
≪ 1
y
n
≈ n
b
N + 1
cos(γt + χ) ,