Chapter 7.3, Problem 9E

Interpretation Introduction

Interpretation:

• a) Approximate the shape of the orbit $\text{r(θ)}$ for $\mu \ll 1$ by assuming a power series solution

  ${\text{r(θ) = 1+μr}}_{\text{1}}\text{(θ)+O}\left({\text{μ}}^{\text{2}}\right)$; substitute this series solution in a differential equation $\frac{\text{dr}}{\text{dθ}}$. By neglecting the higher order terms, derive the simple differential equation ${\text{r}}_{\text{1}}\text{(θ)}$ and solve this differential equation explicitly for ${\text{r}}_{\text{1}}\text{(θ)}$.

• b) To find the maximum and minimum value of r on approximate orbit,- show it lies in the annulus $\sqrt{\text{1-μ}}\text{< r <}\sqrt{\text{1+μ}}$

• c) By using a computer, calculate $\text{r(θ)}$ numerically for small values of $\mu$, and plot the result on the same graph of analytical approximation for $\text{r(θ)}$.

Concept Introduction:

• ➢ Poincare-Bendixson Theorem: If

  • 1) R is closed, subset of the plane is bounded,

  • 2) $\dot{\text{x}}\text{=f(x)}$ is continuously differentiable vector field on an open set containing R

  • 3) R doesn’t have any fixed points

  There exists a trajectory C. The trajectory C is confined in R, in the sense that it starts in R and stays in R for all future time. Either C is closed orbit, or it spirals towards a closed orbit as $t\to \infty$. In any case, R contains a closed orbit