Chapter 6.5, Problem 10E

Interpretation Introduction

Interpretation:

To sketch the phase portrait in the $\left(\text{r, p}\right)$ plane, if $\text{k > 0}$ for a particle moving under the influence of an inverse square force governed by the Hamiltonian $\text{H}\left(\text{p, r}\right)\text{ = }\frac{{\text{p}}^{\text{2}}}{\text{2}}\text{ + }\frac{{\text{h}}^{\text{2}}}{{\text{2r}}^{\text{3}}}\text{ - }\frac{\text{k}}{\text{r}}$. To show that the trajectories are closed if $\frac{{\text{- k}}^{\text{2}}}{{\text{2h}}^{\text{2}}}\text{ < E < 0}$. To find what happens when $\text{E > 0}$ and $\text{E = 0}$. To shows that there are no periodic orbits when $\text{k < 0}$.

Concept Introduction:

Fixed points occur where $\dot{\text{r}}\text{ = 0}$ and $\dot{\text{p}}\text{ = 0}$.

The Jacobian matrix is given by:

$\text{J}\left(\text{r, p}\right)\text{ = }\left(\begin{array}{cc}\frac{\text{d}\dot{\text{r}}}{\text{dr}}& \frac{\text{d}\dot{\text{r}}}{\text{dp}}\\ \frac{\text{dV}}{\text{dr}}& \frac{\text{dV}}{\text{dP}}\end{array}\right)$

The Eigen value $\text{λ}$ can be calculated using the characteristic equation

$\left|\left(\text{A - λI}\right)\right|\text{ = 0}$