Chapter 8.4, Problem 7E

Interpretation Introduction

Interpretation:

To show that the averaged system has no closed orbits.

Concept Introduction:

According to Dulac’s criterion as the value of $\nabla {\text{.x}}^{\text{'}}$ is less than zero, there are no closed orbits for the averaged system.

The averaged equations of the system are,

${\text{r}}^{\text{'}}\text{= -}\frac{\text{1}}{\text{2}}\text{( kr + Fsin}\varphi \text{)}$

$\varphi \text{' = - }\frac{\text{1}}{\text{8}}\left({\text{4a- 3br}}^{\text{2}}\text{+}\frac{\text{4F}}{\text{r}}\text{cos}\varphi \right)$.