How was V(x) derived on slide 13 of Lecture 3-1?

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V(x) is a Lyapunov function. There is no unique rule to find Lyapunov functions (if they exist) for a given system - see Lecture 3-2. For most mechanical or electrical systems it is possible to find Lyapunov functions by writing down a function that stores the total energy of the system, and analyze its derivative along the system trajectories. Energy dissipation usually indicates that the system has an equilibrium point, where the Lyapunov - energy function reaches zero.