The dynamical system of a pendulum is given by $\backslash$theta $=$b$\backslash$theta $$ sin$(\backslash$theta $)$ where $\backslash$theta is the angle

between the position of the pendulum and straight down and b represents a damping force

such as mild air resistance. We investigate the stability of the equilibrium point $\backslash$theta $=\backslash$theta $=0.$

$($a$)$ The system energy $($kinetic plus potential energy$)$ is given by V $(\backslash$theta $,\backslash$theta $)=1$

$2$

$\backslash$theta $2+1$cos$(\backslash$theta $).$

For this part, assume that b $=0.$ Use the Lyapunov direct method $($aka second method$)$

with V as the Lyapunov function to see what the method implies about the senses $($LS$,$

AS$,$ GAS$),$ if any, in which the system is stable. For consistency, use the coordinates x

with x$1=\backslash$theta and x$2=\backslash$theta $.$

$($b$)$ Repeat part $($a$)$ but now assume b $>0.$

$($c$)$ Assuming b $>0$ as in part $($b$),$ examine the linear dynamical system obtained by linearizing

the dynamics around the equilibrium point xe $=.$ What can you conclude

about the stability of xe for the original system $($with b $>0)$ from properties of the

linear system?