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Spherical pendulum

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Spherical pendulum

In physics, a spherical pendulum is a higher dimensional analogue of the pendulum. It consists of a mass m moving without friction on the surface of a sphere. The only forces acting on the mass are the reaction from the sphere and gravity.

Contents

Owing to the spherical geometry of the problem, spherical coordinates are used to describe the position of the mass in terms of (r, θ, φ), where r is fixed. In what follows l is the constant length of the pendulum, so r = l.

Lagrangian mechanics

The Lagrangian is

L = 1 2 m r 2 ( θ ˙ 2 + sin 2 ⁡ θ   ϕ ˙ 2 ) + m g r cos ⁡ θ .

The Euler–Lagrange equations give :

d d t ( m r 2 θ ˙ ) − m r 2 sin ⁡ θ cos ⁡ θ ϕ ˙ 2 + m g r sin ⁡ θ = 0

and

d d t ( m r 2 sin 2 ⁡ θ ϕ ˙ ) = 0

showing that angular momentum is conserved.

Hamiltonian mechanics

The Hamiltonian is

H = P θ θ ˙ + P ϕ ϕ ˙ − L

where

P θ = ∂ L ∂ θ ˙ = m r 2 θ ˙

and

P ϕ = ∂ L ∂ ϕ ˙ = m r 2 ϕ ˙ sin 2 ⁡ θ

References

Spherical pendulum Wikipedia
(Text) CC BY-SA