Chapter 6.7, Problem 4E

Interpretation Introduction

Interpretation:

For a pendulum system governed by $\ddot{\text{θ}}\text{ + sinθ = 0}$, using conservation of energy ${\dot{\text{θ}}}^{\text{2}}\text{ = 2(cosθ - cosα)}$ and $\text{T = 4}{\displaystyle \underset{\text{0}}{\overset{\text{α}}{\int }}\frac{\text{dθ}}{\sqrt{\text{2}\left(\text{cosθ - cosα}\right)}}}$ as well as using the half-angle formula $\text{T = 4}{\displaystyle \underset{\text{0}}{\overset{\text{α}}{\int }}\frac{\text{dθ}}{\sqrt{\text{4}\left({\text{ sin}}^{\text{2}}\left(\frac{\text{α}}{\text{2}}\right){\text{ - sin}}^{\text{2}}\left(\frac{\text{θ}}{\text{2}}\right)\right)}}}$ is to be proved. Also, $\text{T = 4}{\displaystyle \underset{\text{0}}{\overset{\frac{\text{π}}{\text{2}}}{\int }}\frac{\text{d}\varphi }{\text{cos}\left(\frac{\text{θ}}{\text{2}}\right)}}\text{ = 4K}\left({\text{sin}}^2\frac{1}{2}\alpha \right)$ and $\text{T = 2π}\left(\text{1 + }\frac{\text{1}}{\text{16}}{\text{α}}^{\text{2}}\text{ + O}\left({\alpha }^{\text{4}}\right)\right)$ for $\text{α << 1}$ is to be derived.

Concept Introduction:

The pendulum is an example of nonlinear system. It is expressed by a differential equation $\ddot{\text{θ}}\text{ + sinθ = 0}\text{.}$

This system is reversible as well as conservative.

Trigonometric identity:

${\text{cosθ = cos}}^{\text{2}}\left(\frac{\text{θ}}{\text{2}}\right){\text{ - sin}}^{\text{2}}\left(\frac{\text{θ}}{\text{2}}\right)$