| ( C + Ma2 + A | a | cos θ ) ψ̇2 = Mg | a2 | cot θ, |
| c | c |
(4)
where use has been made of the obvious relation nα = cψ̇. If c and θ be given this formula determines the value of ψ̇ for which the motion will be steady.
In the case of the top, the equation of energy and the condition of constant angular momentum (μ) about the vertical OZ are sufficient to determine the motion of the axis. Thus, we have
1⁄2A (θ̇2 + sin2 θψ̇2) + 1⁄2Cn2 + Mgh cos θ = const.,
(5)
A sin2 θψ̇ + ν cos θ = μ,
(6)
where ν is written for Cn. From these ψ̇ may be eliminated, and on differentiating the resulting equation with respect to t we obtain
| Aθ̈ − | (μ − ν cos θ) (μ cos θ − ν) | − Mgh sin θ = 0. |
| A sin3 θ |