How do I prove that the given system is globally asymptotically stable, using Lyapunov analysis?

Question

How do I prove that the given system is globally asymptotically stable, using Lyapunov analysis?

\begin{equation} \left.\begin{aligned} \dot{x_1} &= x_2 \\ \dot{x_2} &= -\frac{x_1}{1 + x_2^2} \label{eq:q2} \end{aligned}\qquad\right\} \end{equation}

Answer 1

You could try to get the Lyapunov function in the form $$ V=\frac12x_1^2+g(x_2) $$ so that $$ \dot V=x_1x_2+g'(x_2)\frac{-x_1}{1+x_2^2} $$ so that one would get a usable result with $g'(x_2)=x_2(1+x_2^2)$, integrating to,for example, $g(x_2)=\frac14(1+x_2^2)^2$.