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this shows that the variation of χ is simple-harmonic, with the period
2π / √(1 + 3 cos2 α)·Ω
As regards the most general motion of a spherical pendulum, it is obvious that a particle moving under gravity on a smooth sphere cannot pass through the highest or lowest point unless it describes a vertical circle. In all other cases there must be an upper and a lower limit to the altitude. Again, a vertical plane passing through O and a point where the motion is horizontal is evidently a plane of symmetry as regards the path. Hence the path will be confined between two horizontal circles which it touches alternately, and the direction of motion is never horizontal except at these circles. In the case of disturbed steady motion, just considered, these circles are nearly coincident. When both are near the lowest point the horizontal projection of the path is approximately an ellipse, as shown in § 13; a closer investigation shows that the ellipse is to be regarded as revolving about its centre with the angular velocity 2⁄3 abΩ/l2, where a, b are the semi-axes.
To apply the equations (11) to the case of the top we start with the expression (15) of § 21 for the kinetic energy, the simplified form (1) of § 20 being for the present purpose inadmissible, since it is essential that the generalized co-ordinates employed should be competent to specify the position of every particle. If λ, μ, ν be the components of momentum, we have
| λ = ∂T / ∂θ̇ | = Aθ̇, |
| μ = ∂T / ∂ψ̇ | = A sin2 θψ̇ + C (φ̇ + cos θψ̇) cos θ, |
| ν = ∂T / ∂φ̇ | = C (θ̇ + cos θψ̇). |
(27)
The meaning of these quantities is easily recognized; thus λ is the angular momentum about a horizontal axis normal to the plane of θ, μ is the angular momentum about the vertical OZ, and ν is the angular momentum about the axis of symmetry. If M be the total mass, the potential energy is V = Mgh cos θ, if OZ be drawn vertically upwards. Hence the equations (11) become
| Aθ̇ − A sin θ cos θψ̇2 + C (φ̇ + cos θψ̇) ψ̇ sin θ = Mgh sin θ, |
| d/dt · { A sin2 θψ̇ + C(φ̇ + cos θψ̇) cos θ } = 0, |
| d/dt · { C (φ̇ + cos θψ̇) } = 0, |
(28)