III.
.
. , . , . .
. , . , m l (.).
E = T + U. , .
dr dr. . , , . , dU = m × g × sin a dr.
:
dt, mv = mdr / dt, :
. (3.1)
a, dr = l d a
. (3.2)
, . , , , , a << 1. sin a ~ a, :
. (3.3)
(3.3) ( )
a = a0cos(ωt+j0), (3.4)
a0 , ; ω , ω=2p/ T; j0 , (a0 cos j0) (t = 0).
(3.4) (3.3), , :
, (3.5)
. , :
. (3.6)
, , .
(.). m k, x F = k×x. :
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, (3.7)
(5.3):
. (3.8)
:
, (3.9)
, (3.4):
x (t) = xmcos (ω0 t +a0). (3.10)
. x(t) y(t) (.):
x (t)= Rcos (ω t +a), (3.11)
y (t) = Rsin (ω t + a) = Rcos (ω t +ap/2),
ω= v / R v. , y , x, p/2.
. a(t), x (t), x (t) y (t) ( ). . , , , .
, , ( oscillation ). , cos (ω t) ω, ( ).
, :
, (3.12)
:
x (t)= Acos (ω0 t +a), (3.13)
A ; ω0 ; ω0 t +a .
, , .
. , . , , x (t)= Acos (ω0 t +a), v = Aω0sin (ω0 t +a), :
.
k :
.
. (3.14)
, . : T = J ω2/2, J . mA 2.
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z
z = x + iy, (1)
x y , i (i 2=1). x z. x = Rez. z (: y = lmz).
z *= x iy. (2)
x + iy. x x.
z , x, y (.). z. , x y . ρ φ.
x = ρ∙ cos φ, y = ρ∙ sin φ, , φ= arctg (y / x). (3)
, z, ( | z |). ,
z= .
φ z.
(3), :
z =ρ(cos φ+ i sin φ).
z 1= x 1+ iy 1 z 2= x 2+ iy 2 , :
z 1= z 2, x 1= x 2 y 1= y 2.
, , 2π:
ρ1 = ρ2, φ1=φ22kπ.
(1) (2) , , z *= z, z , . . z . , z
z * = z.
ei φ = s φ + isin φ, (4)
. φ φ , cos (φ)= cos φ, a sin (‑φ) = sin φ,
e ‑ i φ = s φ ‑ i ∙ sin φ. (5)
(4) (5) cosφ.
s φ = 1/2∙(ei φ +‑ i φ).
(5) (4), , sin φ = (1/2 i) (ei φ ‑ e ‑ i φ).
(4) :
z = ρ e ‑ i φ.
z * = ρ e ‑ i φ.
:
z 1+ z 2=(x 1+ x 2)+ i (y 1+ y 2).
, :
z = z 1∙ z 2 = ρ1 ei φ1∙ρ2 ei φ2 = ρ1ρ2 ei (φ1 + φ2)
, :
ρ=ρ1∙ρ2, φ=φ1+φ2.
:
,
z ∙ z * = ρ2.
( ).
, . , . ω0:
x 1= A 1 cos (ω0 t +a1) x 2= A 2 cos (ω0 t +a2).
x (t) x = x 1 + x 2. x 1 x 2 (.), , x. x 1 x 2, , . , x = x 1 + x 2, x 1 x 2. , x 1, x 2, x A 1 cos (ω0 t +a1) 2 cos (ω0 t +a2), x . : x (t)= x 1+ x 2= = Acos (ω0 t +a). , . . . .
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, (3.15)
:
. (3.16)
(3.15) , . a1a2=0, , A 1 A 2 A = A 1 + A 2. p, , .. A = | A 1 A 2|.
, . , x1 x2 (.). .
. x y, (. (3.11)). , , x y. .
, . . : . , , .
, . , , , . , . , .
.
, . , . , , F * :
. (3.18)
r , . , F * v ; , x .
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. (3.19)
(3.20)
ω0 ‑ , ( r = 0). .
(3.19) :
. (3.21)
(3.21) x = e λ t
(3.22)
, . (3.23)
( β<ω0) . (iω)2, ω ,
. (3.24)
:
, . (3.25)
(58.1)
.
, (3.21)
. (3.26)
a0 α , ω , (3.24). . (3.26). , x.
(3.26) ω , a (t) = a 0 e ‑β∙ t . . a (t), a 0 . x 0 , a 0, α: x 0 = a 0∙cosα.
β = r /2 m, . τ, e . e ‑β∙τ = e ‑1, β∙τ = 1. , , e .
(3.24)
. (3.27)
() T 0 = 2π/ω0. .
- (, a ', a '', a ''' .. . . , a ' = a 0 e ‑β∙ t , a '' = a 0 e ‑β(t + T) = a ' e ‑β T , a ''' = a 0 e ‑β(t +2 T) = a '' e ‑β T . . , , , ,
.
, :
(3.28)
( λ (3.23) (3.25)!).
λ. (3.28) β λ, T,
.
τ, , Ne = τ/ T . , . , , , e .
, (3.29)
. , Ne, τ, e .
(58.7) E = kx 2/2 + mv 2/2
, (3.30)
y = arctg (β/ω). . . , , . ,
.
, E (t), , t. dE / dt < 0.
(β<<ω0) , , (3.30) ,
E = E 0 e ‑2β t , (3.31)
E 0 = k (a 0)2/2 . , (3.30) E (t) t T /2 t + T /2 (T ), , (2β t) T .
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(3.27) , . β=ω0 , . . .
β>ω0 (. (3.25)) (3.21) :
.
C 1 C 2 , ( x 0 v 0)., () , .
. . , . , 2, , , x 0, v 0
. (3.32)
, . , , (. . v 0=0) (, v 0 (3.32)), 1 .