Сягло И.С. Теоретическая механика
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|a − b| ≤ a
1
(t) ≤ a + b.
a b
λ 6= 0
−π < δ < 0
λ
2
> ω
2
o
/2
γ
o
=
q
ω
2
o
− 2λ
2
λ ¿ ω
o
λ γ
o
ω
o
γ = ω
o
+ ε ε ¿ ω
o
λ ε
b =
f
o
2µω
o
√
ε
2
+ λ
2
, tgδ =
λ
ε
.
π/2
ε
ε
1
ε
1
= λ
λ Q
Q =
b(γ = ω
o
)
b(γ = 0)
=
ω
o
2λ
=
ω
o
2ε
1
,
ε
1
b
2
=
f
2
o
4µ
2
ω
2
o
λ
2
1
2
b
2
b
2
ε
ε
1
ε
1
q
io
x
i
= q
i
− q
i0
, ˙x
i
= ˙q
i
.
U(q
i
) = U(q
io
+ x
i
) ≈ U(q
io
) +
1
2
s
X
i=1
s
X
j=1
k
ij
x
i
x
j
.
s
k
ij
=
∂
2
U
∂q
i
∂q
j
¯
¯
¯
¯
¯
¯
¯
q
i
=q
io
.
k
ij
i, j
U(q
io
)
x
i
k
ij
a
ij
(q
i
)
˙
x
i
a
ij
(q
i
) ≈ a
ij
(q
io
) = µ
ij
, T ≈
1
2
s
X
i=1
s
X
j=1
µ
ij
˙
x
i
˙
x
j
.
µ
ij
µ
ij
k
ij
U(q
io
)
L =
1
2
s
X
i=1
s
X
j=1
µ
ij
˙x
i
˙x
j
−
1
2
s
X
i=1
s
X
j=1
k
ij
x
i
x
j
.
x
i
˙x
i
k
ij
µ
ij
x
l
˙x
l
∂L
∂x
l
=
s
X
j=1
k
l j
x
j
,
∂L
∂ ˙x
l
=
s
X
j=1
µ
l j
˙x
j
.
s
X
j=1
(µ
lj
¨x
j
+ k
lj
x
j
) = 0.
x
j
= A
j
e
iωt
,
A
j
ω
x
j
A
j
s
X
j=1
(−ω
2
µ
l j
+ k
lj
)A
j
= 0.
A
j
detk−ω
2
µ
l j
+ k
lj
k = 0
ω
s µ
ij
k
ij
s
s
ω
2
s
ω
2
α
ω
2
α
µ
ij
k
ij
ω
α
ω
α
A
j
A
iα
s
ω
α
s
A
iα
α = 1, 2, . . . , s)
α β
− ω
2
α
s
X
j=1
µ
ij
A
jα
=
s
X
j=1
k
ij
A
jα
, −ω
2
β
s
X
j=1
µ
ij
A
jβ
=
s
X
j=1
k
ij
A
jβ
.
A
iβ
A
iα
i
(ω
2
α
− ω
2
β
)
s
X
i=1
s
X
j=1
µ
ij
A
iα
A
jβ
= 0.
ω
α
6= ω
β
ω
α
= ω
β
α 6= β
α = β)
µ
ij
Ai
α
s
X
i=1
s
X
j=1
µ
ij
A
iα
A
jβ
= δ
αβ
, δ
αβ
=
1, α = β
0, α 6= β
.
A
iα
A
iα
, A
iβ
s
X
i=1
s
X
j=1
k
ij
A
iα
A
jβ
= ω
2
α
δ
αβ
.
k
ij
ω
2
α
A
iα
s
x
iα
= A
iα
e
iω
α
t
.
x
i
=
s
X
α=1
C
α
x
iα
=
s
X
α=1
C
α
A
iα
e
iω
α
t
.
C
α
C
α
= a
α
exp(iϕ
α
)
x
i
=
s
X
α=1
A
iα
a
α
cos(ω
α
t + ϕ
α
).
a
α
ϕ
α
A
iα
Θ
α
x
i
=
s
X
α=1
A
iα
Θ
α
.
x
i
Θ
α
A
iα
T =
1
2
s
X
i=1
s
X
j=1
µ
ij
(
s
X
α=1
A
iα
˙
Θ
α
)(
s
X
β=1
A
jβ
˙
Θ
β
) =
1
2
s
X
α=1
˙
Θ
2
α
,
U =
1
2
s
X
i=1
s
X
j=1
k
ij
(
s
X
α=1
A
iα
Θ
α
)(
s
X
β=1
A
jβ
Θ
β
) =
1
2
s
X
α=1
ω
2
α
Θ
2
α
,
L =
1
2
s
X
α=1
˙
Θ
2
α
−
1
2
s
X
α=1
ω
2
α
Θ
2
α
.