In Example (1.21), investigate the stability of the two additional equilibrium points in the case \(\omega^{2} \sin \alpha>g / a\).

We are given an inequality \(\omega^{2} \sin \alpha > g / a\), and the task is to investigate the stability of the additional equilibrium points. This would require us to identify the system's equations of motion and the equilibrium points first.

For this step, further information on the exact physical system and its corresponding equations of motion are needed. Based on the mention of \(\omega\), it is likely to involve pendulum motion, and a complete set of equations of motion can be obtained from considering kinetic and potential energy expressions. The identified equilibrium points will depend on the specific system and given parameters. For example, assuming a pendulum, there might be equilibrium points at the bottom and top of the motion.

Once we have the equations of motion for the given problem, we will need to linearize them near the equilibrium points. That is, we will find out how the equations of motion behave when they are slightly perturbed from their equilibrium points. We can use the small-angle approximation and Taylor series expansion for this purpose.

After linearizing the equations of motion, we can now study their stability. We can analyze the behavior of the linearized equations by studying the Jacobian by evaluating the Jacobian matrix at the equilibrium points. Then, we can determine its eigenvalues, which will help us to infer the stability of the equilibrium points.

Using the eigenvalues found in Step 4, we can analyze the stability of the equilibrium points. If the real part of all eigenvalues is negative, the equilibrium point is stable (attracting). If the real part of one or more eigenvalues is positive, the equilibrium point is unstable (repelling). If there are complex eigenvalues, their phases can give insights into the oscillatory nature of the system (e.g., stable spiral or unstable spiral). In conclusion, to investigate the stability of the two additional equilibrium points as given in Example (1.21), follow these five steps: 1. Identify the problem 2. Determine the equilibrium points and equations of motion 3. Linearize the equations of motion near equilibrium points 4. Study the behavior of the linearized equations of motion. 5. Analyze the stability of the system.