Question 11.Q.4: Form of the motion This is all very well, but what does the ...
Form of the motion
This is all very well, but what does the motion actually look like?
Despite the problem being called integrable, the integrals arising from the separation procedure cannot be evaluated and no explicit solution is possible. However, equation (11.19) has the form of an energy equation for a system with one degree of freedom.
We have met this situation before with the radial motion equation in orbit theory and the deductions we can make are the same. Because the left side of (11.19) is positive, it follows that the motion is restricted to those values of θ that make the function
\dot{\theta}^{2}=\frac{u^{2}}{a^{2}}(\cos \alpha-\cos \theta)\left(\frac{\cos \alpha+\cos \theta}{\sin ^{2} \theta}-\frac{2 a g}{u^{2}}\right) (11.19)
F=(\cos \alpha-\cos \theta)\left(\frac{\cos \alpha+\cos \theta}{\sin ^{2} \theta}-\frac{2 a g}{u^{2}}\right)
positive. Moreover, maximium and minimum values of θ can only occur when F(θ ) = 0.
Since F(α) = 0, θ = α is one extremum* and any other extremum must be a root of the equation G(θ ) = 0, where
G=\frac{\cos \alpha+\cos \theta}{\sin ^{2} \theta}-\frac{2 a g}{u^{2}}.
Whether α is a maximum or minimum point of θ depends on the value of the initial projection speed u. On differentiating equation (11.19) with respect to t, we find that the initial value of \ddot{\theta} is given by
\left.\ddot{\theta}\right|_{\theta=\alpha}=\frac{u^{2}}{a^{2}}\left(\frac{\cos \alpha}{\sin ^{2} \alpha}-\frac{a g}{u^{2}}\right),
so that θ initially increases if u²/ag > sin² α/ cos α, and θ initially decreases if u²/ag < sin² α/ cos α. (The critical case corresponds to the special case of conical motion.) Suppose that the first condition holds. Then \theta_{\min }=\alpha \text { and } \theta_{\max } must be a root of the equation G(θ ) = 0. Since G(α) > 0 and G(π − α) < 0, such a root does exist and is less than π − α.
For example, consider the particular case in which α = π/3 and u² = 4ag. Then the equation G(θ ) = 0 simplifies to give
\cos \theta(\cos \theta+2)=0,
from which it follows that \theta_{\max }=\pi / 2 . Hence, in this case, θ oscillates periodically in the range π/3 ≤ θ ≤ π/2.
At the same time as the coordinate θ oscillates, the coordinate Φ increases in accordance with equation (11.18). Hence, during each oscillation period τ of the coordinate θ, Φ increases by
\dot{\phi}=\frac{u \sin \alpha}{a \sin ^{2} \theta} (11.18)
\Phi=\frac{u \sin \alpha}{a} \int_{0}^{\tau} \frac{d t}{\sin ^{2} \theta}.
This pattern of motion repeats itself with period τ , but the motion is only truely periodic if it eventually links up with itself; this occurs only when the initial conditions are such that Φ/π is a rational number.
Figure 11.8 shows an actual path of the spherical pendulum, corresponding to the initial conditions α = π/6 and u²/g = 1.9. The results are entirely consistent with the theory above.
