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7 Lagrangian and Hamiltonian Mechanics
Fig. 7.19
T
1
2
m 1 l
2
1
˙
θ
2
1 +
1
2
m 2
l
2
1
˙
θ
2
1 + l
2
2
˙
θ
2
2 + 2l 1 l 2 ˙
θ 1 ˙
θ 2
(4)
V = m 1 gl 1 (1 − cos θ 1 ) + m 2 gl 1 (1 − cos θ 1 ) + m 2 gl 2 (1 − cos θ 2 )
m 1 gl 1
θ 2
1
2
+
m 2 g
2
l 1 θ
2
1 + l 2 θ
2
2
(5)
L =
1
2
m 1 l
2
1
˙
θ
2
1 +
1
2
m 2
l
2
1
˙
θ
2
1 + l
2
2
˙
θ
2
2 + 2l 1 l 2 ˙
θ 1 ˙
θ 2
− m 1 gl 1
θ 2
1
2
−
m 2 g
2
l 1 θ
2
1 + l 2 θ
2
2
(6)
∂ L
∂ ˙
θ 1
= m 1 l
2
1
˙
θ 1 + m 2 l
2
1
˙
θ 1 + m 2 l 1 l 2 ˙
θ 2
(7)
∂ L
∂θ 1
= −m 1 gl 1 θ 1 − m 2 gl 1 θ 1 = −(m 1 + m 2 )gl 1 θ 1
(8)
∂ L
∂ ˙
θ 2
= m 2 l
2
2
˙
θ 2 + m 2 l 1 l 2 ˙
θ 1
(9)
∂ L
∂θ 2
= −m 2 gl 2 θ 2
(10)
Lagrange’s equations are
d
dt
∂ L
∂ ˙
θ 1
−
∂ L
∂θ 1
= 0,
d
dt
∂ L
∂ ˙
θ 2
−
∂ L
∂θ 2
= 0,
(11)
using (7, (8), (9) and 10) in (11) we obtain the equations of motion
(m 1 + m 2 )l 1 ¨
θ 1 + m 2 l 2 ¨
θ 2 + (m 1 + m 2 )gθ 1 = 0
(12)
l 2 ¨
θ 2 + gθ 2 + l 1 ¨
θ 1 = 0
(13)