Сягло И.С. Теоретическая механика
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dU
dq
¯
¯
¯
¯
¯
¯
q=q
o
= 0,
d
2
U
dq
2
¯
¯
¯
¯
¯
¯
¯
q=q
o
= k > 0.
d
2
U/dq
2
k
x
x = q − q
o
, ˙x = ˙q.
a(q) U(q)
a(q) U(q)
T =
1
2
a(q) ˙q
2
≈
1
2
a(q
o
) ˙q
2
=
1
2
µ ˙x
2
, U(q) ≈ U(q
o
) +
1
2
kx
2
.
µ = a(q
o
)
a(q)
U(q
o
)
L =
1
2
µ ˙x
2
−
1
2
kx
2
.
µ k
Φ =
1
2
α
o
˙
x
2
, α
o
= α(q
o
).
d
dt
∂L
∂ ˙x
−
∂L
∂x
= −
∂Φ
∂ ˙x
.
µ¨x + kx = −α
o
˙x.
µ
ω
2
o
=
k
µ
, 2λ =
α
o
µ
.
¨
x + 2λ
˙
x + ω
2
o
x = 0.
x = exp (νt)
ν
1
= −λ +
r
λ
2
− ω
2
o
, ν
2
= −λ −
r
λ
2
− ω
2
o
.
λ ≥ ω
o
x = C
1
e
−(λ+
√
λ
2
−ω
2
o
)t
+ C
2
e
−(λ−
√
λ
2
−ω
2
o
)t
, ν
1
6= ν
2
,
x = (C
1
+ C
2
t)e
−λt
, ν
1
= ν
2
.
C
1
C
2
x
λ < ω
o
ω =
r
ω
2
o
− λ
2
.
x = e
−λt
(A cos ωt + B sin ωt), x = ae
−λt
cos(ωt + β).
a =
√
A
2
+ B
2
, tgβ = −
B
A
.
λ = 0 ω = ω
o
x = A cos ωt + B sin ωt, x = a cos(ωt + β).
ω
a β
E =
1
2
m ˙x
2
+
1
2
kx
2
=
1
2
mω
2
a
2
.
A exp(iω) A
A = a exp(iβ)
Ae
iωt
= ae
i(ωt+β)
= a cos(ωt + β) + ia sin(ωt + β),
x = Re[Ae
iωt
] = a cos(ωt + β).
λ
U (q, t)
U (q, t) ≈ U (0, t) +
∂U (q, t)
∂q
¯
¯
¯
¯
¯
¯
¯
q=q
o
x = U (0, t) −f(t) x,
(−f(t))
U (0, t)
L =
1
2
m ˙x
2
−
1
2
kx
2
+ f(t) x.
¨x + 2λ ˙x + ω
2
o
x =
1
µ
f(t).
x = ae
−λt
cos(ωt + β) + ˜x.
f(t) = f
o
cos γt = Re[f
o
e
iγt
],
f
o
¨x + 2λ ˙x + ω
2
o
x =
1
µ
f
o
e
iγt
.
˜x ˜x = B exp(iγt)
˜x
B
B =
f
o
µ(ω
2
o
− γ
2
+ 2iλγ)
=
f
o
(ω
2
o
− γ
2
− 2iλγ)
µ((ω
2
o
− γ
2
)
2
+ 4λ
2
γ
2
)
.
B B = b exp(iδ)
b = |B| =
f
o
µ
r
(ω
2
o
− γ
2
)
2
+ 4λ
2
γ
2
, tgδ = −
2λγ
ω
2
o
− γ
2
.
x = ae
−λt
cos(ωt + β) + b cos(γt + δ).
λ = 0
ω
o
γ
x = a cos(ω
o
t + β) +
f
o
µ(ω
2
o
− γ
2
)
cos γt.
γ → ω
o
γ = ω
o
˜x = Bt exp(iω
o
t) B
B = −
if
o
2ω
o
µ
.
x = a cos(ω
o
t + β) +
f
o
t
2µω
o
sin γt.
π/2
γ = ω
o
+ ε ε ¿ ω
o
x = (a + be
i(εt−β)
)e
i(ω
o
t+β)
.
x = a
1
(t) cos(ω
o
t + δ(t)),
a
1
(t) = |a + be
i(εt−β)
| =
r
a
2
+ b
2
+ 2ab cos(εt − β).
ω
o
a
1
(t) δ(t)
ε