Equilibrium solutions of a spherical pendulum

[NobodyMinus]

Homework Statement

A spherical pendulum consists of bob of mass m attached to a massless rod of fixed length R. The end of the rod opposite the bob pivots freely (in two directions) about some fixed point.

For the conical pendulum (θ=constant) case, show that the conical angle is stable for θ< θ/2. That is, show that if θ=θ₀+ε then oscillates about θ₀ in harmonic motion. Plot the frequency of oscillation for angles 0<θ<∏/2. Comment on any interesting aspects of the curve.

g = 9.8, R = 1.8824, m = 1.96706

Homework Equations

$$H=\frac{1}{2mR^2}\left(p_\theta^2 + \frac{1}{\sin^2\theta} p_\phi^2\right) + mgR(1-\cos\theta)$$

The Attempt at a Solution

I don't really know how to type this out too well, but I took the derivative of H with respect to the canonical momentum in theta to get the first derivative of theta, then took the time derivative of that. this yielded
$$\ddot{\theta} = \frac{\dot{p}_\theta}{mR^2}$$
The hamiltonian to find the first derivative of p_θ and plugging it in yielded
$$\ddot{\theta} = \frac{p_\phi^2}{m^2 R^4 \tan\theta\sin^2\theta}-\frac{g\sin\theta}{R}$$
and was from here unable to find an equilibrium solution

Places where pθ or p$\phi$ appear are meant to indicate p_θ and p_$\phi$ respectively, I just couldn't figure out how to do subscripts in latex
Moderator note: I reformatted your equations using LaTeX. Let me know if I made a mistake.
You guys are my last hope. Thanks!

 

[BruceW]

hi, welcome to physicsforums!
You have done most of the hard work already. The question is asking about the behaviour of theta. So try to write the right-hand-side of your last equation in terms of just theta and constants.

Edit: ah, also, I guess you know that $p_{\phi}$ is a constant. But you will need to write it out explicitly, or finding an equilibrium solution (using the form the equation is currently in) is difficult (as you know already).

Edit again: no, sorry you don't need to write out $p_{\phi}$ explicitly. For some reason, I thought the equation in it's current form is bad. But it is fine in the form it is in now. So, the right-hand side is a function of theta only. And they want you to describe the behaviour close to $\theta_0$ How would you do this? (hint: it is effectively just a 1d problem now, so think of what you would do in a 1d problem).