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A brief working note dated 1.9.17 from the Appendix F dumbbell satellite folder. Phil describes Maple runs with different starting values of φ, where φ jumps by π when the solution passes through θ = 0 (the cot(θ) issue), and asks how to restrict the angle ranges. He compares with a spherical pendulum example and checks periods: with ω = 1, out-of-plane T/2 = 3.14 and in-plane T/1.73 = 3.63, matching (F.5.13) and (F.5.10). Figures are not captured in the text.
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Maple numerical solution of the dumbbell satellite PhL 1.9.17
When I start at φ = π/2, I get the left below
where φ remains at π/2, and θ has both positive and negative values. But when I start at φ = 0 I get the right above where it does not pass through θ = 0 by instead forces φ to jump by π. Is there some way in Maple to restrict the desired range of φ and/or θ?
Here is a guy who thinks he is doing the spherical pendulum, I should see if he agrees with me
Notice that he has that same cot(θ) issue that I do, but he does not bring it up.
This starts with φ = 0 and so is out-of-plane oscillation
With ω = 1 we have T = 6.28 sec.
We know from (F.5.13) that
Toutofplane = Tocs2 = T/2 = 6.28/2 = 3.14 verified in the figure
On the other hand we know from (F.5.10) that
Tomplane = Tocs1 = T/ = 6.28/1.73 = 3.63 verified in the figure