| θ̈ − sin θ cos θψ̇2 | = − (g/l) sin θ, |
| sin2 θψ̇ | = h, |
(23)
where h is a constant. The latter equation expresses that the angular momentum ml2 sin2 θψ̇ about the vertical OZ is constant. By elimination of ψ̇ we obtain
| θ̈ − h2 cos2 θ / sin3 θ = − | g | sin θ. |
| l |
(24)
If the particle describes a horizontal circle of angular radius α with constant angular velocity Ω, we have ω̇ = 0, h = Ω2 sin α, and therefore
| Ω2 = | g | cos α, |
| l |
(25)
as is otherwise evident from the elementary theory of uniform circular motion. To investigate the small oscillations about this state of steady motion we write θ = α + χ in (24) and neglect terms of the second order in χ. We find, after some reductions,
χ̈ + (1 + 3 cos2 α) Ω2χ = 0;