7.3 Solutions
301
The Lagrangian of the system takes the form
L = T − V =
1
2
m 1 ˙
x
2
+
1
2
m 2 (− ˙
x + ˙
x
)
2
+
1
2
m 3 (− ˙
x − ˙
x
)
2
+ (m 1 − m 2 − m 3 )gx + (m 2 − m 3 )gx + m 2 gl + m 3 g(l + l
)
(3)
The equations of motion are then
d
dt
∂ L
∂ ˙
x
−
∂ L
∂ x
= 0
( 4 )
d
dt
∂ L
∂ ˙
x
−
∂ L
∂ x = 0
( 5 )
which yield
(m 1 + m 2 + m 3 ) ¨
x + (m 3 − m 2 ) ¨
x
= (m 1 − m 2 − m 3 )g
(6)
(m 3 − m 2 ) ¨
x + (m 2 + m 3 ) ¨
x
= (m 2 − m 3 )g
(7)
Solving (6) and (7) we obtain the equations of motion.
7.5 T = v
2
˙
u
2
+ 2 ˙
v
2
(1)
V = u
2
− v
2
(2)
L = T − V = v
2
˙
u
2
+ 2 ˙
v
2
− u
2
+ v
2
(3)
∂ L
∂ ˙
u
= 2v
2
˙
u,
∂ L
∂u
= −2u
(4)
∂ L
∂ ˙
v
= 4 ˙
v,
∂ L
∂v
= 2v( ˙
u
2
+ 1)
(5)
The equations of motion
d
dt
∂ L
∂ ˙
u
−
∂ L
∂u
= 0
( 6 )
d
dt
∂ L
∂ ˙
v
−
∂ L
∂v
= 0
( 7 )
yield
2
d
dt
(v
2
˙
u) + 2u = 0
or v
2
¨
u + 2 ˙
u ˙
v + 2u = 0
( 8 )
2 ¨
v + v( ˙
u
2
+ 1) = 0
( 9 )