C P L = C − P
θ1 θ2
m1 m1
m2
L2 L1 θ1
θ2
t m2
m1 x1 (t) = L1 sin(θ1 (t)) y1 (t) = −L1 cos(θ1 (t)) m2 x2 (t) = L1 sin(θ1 (t)) + L2 sin(θ2 (t)) y2 (t) = −L2 cos(θ1 (t)) − L2 cos(θ2 (t))
t
L
C
P
L L xi (t)
F d dt
∂F ∂ x˙i (t)
−
∂F =0 ∂x i (t)
d dt
∂L
−
∂L =0 ∂θ 1 (t)
d dt
−
∂L =0 ∂θ 2 (t)
∂ θ˙1 (t) ∂L ∂ θ˙2 (t)
L
θ˙1
θ˙2
C
C = 12 mv 2
1 1 2 2 (t) + m2 vm (t) = m1 vm 2 2 1 1 = m1 (x˙1 2 (t) + y˙1 2 (t)) + m2 (x˙2 2 (t) + y˙2 2 (t)) 2 2 C =
1
x˙1 (t) x˙2 (t) y˙1 (t) x y
2
y˙2 (t)
x˙1 (t) = L1 cos(θ1 (t))θ˙1 (t) x˙2 (t) = L1 cos(θ1 (t))θ˙1 (t) + L2 cos(θ2 (t))θ˙2 (t) y˙1 (t) = L1 sin(θ1 (t))θ˙1 (t) y˙2 (t) = L1 sin(θ1 (t))θ˙1 (t) + L2 sin(θ2 (t))θ˙2 (t)
x˙1 2 + y˙1 2 x˙1 2 + y˙1 2 = L21 cos2 (θ1 )θ˙12 + L21 sin2 (θ1 )θ˙12 = L21 θ˙12 x˙2 2 + y˙2 2 x˙2 2 + y˙2 2 = L21 cos2 (θ1 )θ˙12 + L22 cos2 (θ2 )θ˙22 + 2L1 L2 cos(θ1 ) cos(θ2 )θ˙1 θ˙2 +
+L21 sin2 (θ1 )θ˙12 + L22 sin2 (θ2 )θ˙22 + 2L1 L2 sin(θ1 )sin(θ2 )θ˙1 θ˙2 = 2 2 = L21 θ˙1 + L22 θ˙2 + 2L1 L2 θ˙1 θ˙2 (cos(θ1 ) cos(θ2 ) + sin(θ1 )sin(θ2 )) = 2 2 = L21 θ˙1 + L22 θ˙2 + 2L1 L2 θ˙1 θ˙2 cos(θ1 − θ2 )
C
C =
1 1 2 2 2 m1 L21 θ˙1 + m2 [L21 θ˙1 + L22 θ˙2 + 2L1 L2 θ˙1 θ˙2 (cos(θ1 − θ2 ))] 2 2
P
P = mgh h
y
P = m1 gy 1 + m2 gy 2 =⇒ P = −m1 gL1 cos(θ1 ) − m2 g (L1 cos(θ1 ) + L2 cos(θ2 ))
P = −(m1 + m2 )gL1 cos(θ1 ) − m2 gL2 cos(θ2 ) L
=
C − P
C −P
L=
1 1 2 2 2 m1 L21 θ˙1 + m2 [L21 θ˙1 + L22 θ˙2 + 2L1 L2 θ˙1 θ˙2 (cos(θ1 − θ2 ))]+ 2 2 +(m1 + m2 )gL 1 cos(θ1 ) + m2 gL2 cos(θ2 ) =
=
1 2˙2 1 2 L1 θ1 (m1 + m2 ) + m2 L22 θ˙2 + m2 L1 L2 θ˙1 θ˙2 (cos(θ1 − θ2 ))+ 2 2 +(m1 + m2 )gL1 cos(θ1 ) + m2 gL2 cos(θ2 )
d dt d dt
− ˙()
∂L
∂ θ1 t ∂L
∂ θ˙2 (t)
−
∂L =0 ∂θ 1 (t) ∂L =0 ∂θ 2 (t)
θ1
∂L =⇒ −m2 L1 L2 θ˙1 θ˙2 (sin(θ1 − θ2 )) − (m1 + m2 )gL1 sin(θ1 ) ∂θ 1 (t) ∂L ∂ θ˙1 (t) d dt
=⇒ L21 θ˙1 (m1 + m2 ) + m2 L1 L2 θ˙2 cos(θ1 − θ2 )
∂L
=⇒ L21 (m1 + m2 )θ¨1 + m2 L1 L2 θ¨2 cos(θ1 − θ2 )− ∂ θ˙1 (t) −m2 L1 L2 θ˙2 sin(θ1 − θ2 )(θ˙1 + θ˙2 )
d dt
∂L
−
∂L = 0 =⇒ L21 (m1 + m2 )θ¨1 + m2 L1 L2 θ¨2 cos(θ1 − θ2 ) ∂θ 1 (t)
∂ θ˙1 (t) −m2 L1 L2 θ˙1 sin(θ1 − θ2 )(θ˙1 + θ˙2 ) + m2 L1 L2 θ˙1 θ˙2 (sin(θ1 − θ2 ))+ +(m1 + m2 )gL1 sin(θ1 ) = 0
L1 (m1 + m2 )θ¨1 + m2 L2 θ¨2 cos(θ1 − θ2 ) + ( m1 + m2 )g sin(θ1 )+ 2 +m2 L2 θ˙2 sin(θ1 − θ2 ) = 0
θ2
∂L =⇒ m2 L1 L2 θ˙1 θ˙2 (sin(θ1 − θ2 )) − m2 gL2 sin(θ2 ) ∂θ 2 (t) ∂L ∂ θ˙2 (t)
=⇒ m2 L22 θ˙2 + m2 L1 L2 θ˙1 cos(θ1 − θ2 )
d dt
∂L
d dt
∂L
=⇒ m2 L22 θ¨2 + m2 L1 L2 θ¨1 cos(θ1 − θ2 )− ∂ θ˙2 (t) −m2 L1 L2 θ˙1 sin(θ1 − θ2 )(θ˙1 + θ˙2 )
−
∂L = 0 =⇒ m2 L22 θ¨2 + m2 L1 L2 θ¨1 cos(θ1 − θ2 )− ∂θ 2 (t)
∂ θ˙2 (t) −m2 L1 L2 θ˙1 sin(θ1 − θ2 )(θ˙1 + θ˙2 ) − m2 L1 L2 θ˙1 θ˙2 (sin(θ1 − θ2 ))+ m2 gL2 sin(θ2 ) = 0
2
L2 θ¨2 + L1 θ¨1 cos(θ1 − θ2 ) + g sin(θ2 ) − L1 θ˙1 sin(θ1 − θ2 ) = 0
θ¨1
θ¨1 =
θ¨2
−m2 L2 θ¨2 cos(θ1 − θ2 ) − (m1 + m2 )g sin(θ1 ) − L1 (m1 + m2 ) 2
m2 L2 θ˙2 sin(θ1 − θ2 ) − L1 (m1 + m2 ) 2 −L1 θ¨1 cos(θ1 − θ2 ) − g sin(θ2 ) + L1 θ˙1 sin(θ1 − θ2 ) ¨ θ2 =
L2
θ¨2
θ¨1
2 −m2 L1 θ˙1 sin(θ1 − θ2 ) cos(θ1 − θ2 ) + gm2 sin(θ2 ) cos(θ1 − θ2 ) ¨ + θ1 = L1 (m1 + m2 ) − m2 L1 cos2 (θ1 − θ2 ) 2 −m2 L2 θ˙2 sin(θ1 − θ2 ) − (m1 + m2 )g sin(θ1 ) + L1 (m1 + m2 ) − m2 L1 cos2 (θ1 − θ2 )
2
m2 L2 θ˙2 sin(θ1 − θ2 ) cos(θ1 − θ2 ) + g (m1 + m2 )sin(θ1 ) cos(θ1 − θ2 ) + θ¨2 = L2 (m1 + m2 ) − m2 L2 cos2 (θ1 − θ2 ) 2
L1 (m1 + m2 )θ˙1 sin(θ1 − θ2 ) − (m1 + m2 )g sin(θ2 ) + L2 (m1 + m2 ) − m2 L2 cos2 (θ1 − θ2 )
z1 z2 z3 z4
= θ1 = θ2 = θ˙1 = θ˙2
=⇒
z˙1 z˙2 z˙3 z˙4
= θ˙1 = θ˙2 = θ¨1 = θ¨2
z˙1 = z3 z˙2 = z4 z˙3 =
−m2 L1 z32 sin(z1
− z2 ) cos(z1 − z2 ) + gm2 sin(z2 ) cos(z1 − z2 ) + L1 (m1 + m2 ) − m2 L1 cos2 (z1 − z2 )
+ z˙4 =
−m2 L2 z42 sin(z1 − z2 ) − (m1 + m2 )g sin(z1 ) L1 (m1 + m2 ) − m2 L1 cos2 (z1 − z2 )
m2 L2 z42 sin(z1 − z2 ) cos(z1 − z2 ) + g (m1 + m2 ) sin(z1 ) cos(z1 − z2 ) + L2 (m1 + m2 ) − m2 L2 cos2 (z1 − z2 )
+
L1 (m1 + m2 )z32 sin(z1 − z2 ) − (m1 + m2 )g sin(z2 ) L2 (m1 + m2 ) − m2 L2 cos2 (z1 − z2 )
m2 m1
L1 m1 = 2 L1 = 1
z1 (0) = π2 z3 (0) = 0
m2 = 1 L2 = 2 z2 (0) = π z4 (0) = 0
m1
m2 m1 = 1 L1 = 1 .5
z1 (0) = 32π z3 (0) = 0
m2 = 1 L2 = 1 z2 (0) = 32π z4 (0) = 0
(X, φ) φ : R × X −→ X
X
φ φ(0, x) = x
x ∈ X
φ(t, φ(s, x)) = φ(t + s, x)
t, s ∈ R X =
x ∈ X R
φ
φx (t)
x x
(X, φ)
∀ε > 0
∃δ > 0 | (||φx (0) − φy (0)|| < δ =⇒ || φx (t) − φy (t)|| < ε
φx (t)
φy (t)
∀t ≥ 0)
t
(X, φ) Λ ⊂ Rn
U t0
V
Λ Λ
φ(t0 , U ) ⊂ V
λ(v) λ(v) = lim
n→+∞
v
1 n
log(||dφn (t, x)v||)
n
R
λ(x0 )
{xn }n
φx (t) 0
λ(x0 ) = lim
n→+∞
1 n
n−1
log( j =0
φxj (t))
φ
x0
λ(v) > 0
ni
s
||dφni (t, x)v || ≥ e(λ(v )−s)ni ||v || 1 (λ(v)−s)ni e d(x, y ) 2
d(φni (t, x), φni (t, y )) ≥
x
y
λ(x0 ) > 0
ni
s n−1
φj (t) ≥ e(λ(x
) s)ni
0 −
j =0
d(φxj , φxk ) ≥
1 (λ(x e 2
) s)ni
0 −
d(xj , xk )
)
x1 = f 1 (x1 , . . . , xn ) xn = f n (x1 , . . . , xn
) =0
f 1 (x1 , . . . , xn ) = 0 f n (x1 , . . . , xn
(0, 0, 0, 0) 0 z3
z4
π
(π, 0, 0, 0) (0, π, 0, 0) (π,π, 0, 0)
(0, 0)
x = Ax A A A
1+ 1+ − − 1 − 1 − −i
g L1
g L1
i
,
m2 m1 +m2
4
,
m2 m1 +m2
g L1
i
m2 m1 +m2
,
m2 m1 +m2
4
(0, 0, 0, 0) z1
g L1
z2
1− 1− 1 − 1 − −i
g L1
i
,
m2 m1 +m2
i
,
4
g L1
m2 m1 +m2
g L1 m2 m1 +m2
g L1
,
4
m2 m1 +m2
θ1
θ2
−
1 − − 1− 4
g L1
m2 m1 +m2
g L1
,
,
1 − 1−
m2 m1 +m2
g L1
−i 4
m2 m1 +m2
g L1 m2 m1 +m2
i ,
1 − 4
,
g L1
m2 m1 +m2
− 1+
,
g L1
1 −
g L1 m2 m1 +m2
4
,
m2 m1 +m2
1+
g L1 m2 m1 +m2
(0, 0, 0, 0) (0, 0, 0, 0) (0, 0)
