
95Lagrangian Formulation of Mechanics
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and the potential energy is
V = mg(l − lcos θ) = mgl(1 − cos θ).
Then the Lagrangian L is
LTVmlmgl=−=−−
1
1
22
(cos )
and the Lagrange’s equation is
d
d
/
t
LL
gl
∂
−
∂
∂
=+ =
θ
()si
.
For a small angle of oscillation, sin θ ≅ θ, the Lagrange’s equation of motion reduces to the
familiar harmonic oscillator type of equation
()gl
.
Example 4.2
A bead slides on a smooth wire that is rotating uniformly about a point O in a horizontal plane (the
xy-plane). Describe the motion of the bead (Figure 4.5).
θ
l
m
O
FIGURE 4.4 Simple pendulum.
Bead
Wire
ω
θ
r
FIGURE 4.5 A bead slides on a ...