Chapter 6.7, Problem 1E

Interpretation Introduction

Interpretation:

To find and classify the fixed points of $\ddot{\text{θ}}\text{ + b}\dot{\text{θ}}\text{ + sinθ = 0}$ for all $\text{b>0}$, and plot the phase portraits for the qualitatively different cases.

Concept Introduction:

Consider the system as,

$\dot{\text{x}}\text{ = f (x, y)}$

And,

$\dot{\text{y}}\text{ = g (x, y)}$.

The linearized system for the above system can be expressed as,

$\left(\begin{array}{l}\dot{\text{x}}\\ \dot{\text{y}}\end{array}\right)\text{ = }\left(\begin{array}{l}\frac{\partial \dot{\text{x}}}{\partial \text{x}}\frac{\partial \dot{\text{x}}}{\partial \text{y}}\\ \frac{\partial \dot{\text{y}}}{\partial \text{x}}\frac{\partial \dot{\text{y}}}{\partial \text{y}}\end{array}\right)\left(\begin{array}{l}\text{x}\\ \text{y}\end{array}\right)$

Here, $\left(\begin{array}{l}\frac{\partial \dot{\text{x}}}{\partial \text{x}}\frac{\partial \dot{\text{x}}}{\partial \text{y}}\\ \frac{\partial \dot{\text{y}}}{\partial \text{x}}\frac{\partial \dot{\text{y}}}{\partial \text{y}}\end{array}\right)$ is called the Jacobian matrix at any fixed point $\left({\text{x}}^{\text{*}}{\text{,y}}^{\text{*}}\right)$.