Student Seminar: Classical and Quantum Integrable Systems

2.3.5 Mathematical pendulum
The theory of elliptic functions finds beautiful applications in many classical problems.
One of them is the motion of the mathematical pendulum in the gravitational
field of the Earth.
Consider the mathematical pendulum (of mass M) in the gravitational field of
the Earth.
L
01
01
M
A pendulum in the gravitational field of the Earth. Here L is its length and G is
the gravitational constant.
G
First we derive the eoms. The radius-vector and the velocity are is
⃗r(t) = (L} sin {{ θ}
, L} cos {{ θ}
) , ⃗v(t) = (L cos θ ˙θ, −L sin θ ˙θ) .
x y
Projecting the Newton equations of the axes x and y we find
Differentiating we get
L d2 cos θ
dt 2 = mg , L d2 sin θ
dt 2 = 0 .
−L(cos θ ˙θ 2 + sin θ¨θ) = mg , − sin θ ˙θ 2 + cos θ¨θ = 0 .
Excluding from these equations ˙θ 2 we obtain the equations of motion
L¨θ = −mg sin θ .
This equation can be integrated once by noting that
i.e. that
d ˙θ 2
dt = 2 ˙θ¨θ = 2 ˙θ ( − mg
L sin θ) = − 2mg
L
sin θ ˙θ = 2mg d
L dt cos θ ,
(
d
˙θ 2 − 2mg )
dt L cos θ = 0 ,
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