Page 199 - Intro to Tensor Calculus

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EXAMPLE 2.2-3. (Simple pendulum) Find the equation of motion for the pendulum system illus-
trated in the figure 2.2-3.
Solution: Choose the angle θ illustrated in the figure 2.2-3 as the generalized coordinate. If the pendulum
is moved from a vertical position through an angle θ, we observe that the mass m moves up a distance
h = ` − ` cosθ. The work done in moving this mass a vertical distance h is

W = −mgh = −mg`(1 − cos θ),

since the force is −mg in this coordinate system. In moving the pendulum through an angle θ, the arc length
s swept out by the mass m is s = `θ. This implies that the kinetic energy can be expressed
2
1 ds 1 2 1 2 ˙ 2
T = m = m `θ ˙ = m` (θ) .
2 dt 2 2

Figure 2.2-3 Simple pendulum system

The Lagrangian of the system is

1 2 ˙ 2
L = T − V = m` (θ) − mg`(1 − cos θ)
2
and from this we find the equation of motion

d ∂L ∂L d 2 ˙
− =0 or m` θ − mg`(− sinθ)=0.
dt ∂θ ˙ ∂θ dt
This in turn simplifies to the equation
g
¨
θ + sin θ =0.
`
˙
This equation together with a set of initial conditions for θ and θ represents the nonlinear differential equation
which describes the motion of a pendulum without damping.