I am studying nonlinear control theory, and I wanted to ask a question about Lyapunov functions please.
So there is the Lyapunov theorem for stability of $\dot{x}=f(x)$ using Lyapunov functions, but the problem is that there are no formulas to determine the Lyapunov functions. Why we can not just study the stability of the point x in the phase space using potential energy ?
I saw some articles where they consider x as a charged particle in space with Lorenz force, but I mean, we can just write : $$\ddot{x} = \frac{df}{dx}\dot{x}$$ and consider this formula as a : 1.acceleration = Forces and then it becomes a study of stability of this point mass ? In this case we could directly use the potential energy of the mass from the right hand side (the forces) to find if it is stable or not no ? Of course i understand that we can not do this for any system since we need for example f to be differentiable, but I don’t understand why we do not just use this potential energy for Lyapunov function ?
Thank you in advance for your answers Best regards