Chapter 8.6, Problem 7E

Interpretation Introduction

Interpretation:

• (a) To show that the system has the solution given by,

  ${\text{r = r}}_{\text{0}}\text{, }\dot{\text{θ}}{\text{= ω}}_{\text{θ}}$ Corresponding to UCM and to find the formulas for ${\text{r}}_{\text{0}}{\text{ and ω}}_{\text{θ}}$

• (b) To find the frequency ${\text{ω}}_r$ of small oscillation about the circular orbit.

• (c) To show that the small radial oscillations correspond to quasiperiodic motion by calculating the winding number $\frac{{\text{ω}}_{\text{r}}}{{\text{ω}}_{\text{θ}}}$

• (d) To show by a geometric argument that the motion is either periodic or quasiperiodic for any amplitude of radial oscillations.

• (e) Give the mechanical realization of the system.

Concept Introduction:

• ➢ The fixed point of a differential equation is a point where, $\text{f(x*) = 0 ; while substitution f(x*) = }\dot{\text{x}}\text{ is used and x\&*\#x00A0;is a fixed point}$.

• ➢ The winding number determines the nature of the trajectory. In case of rational winding number, the trajectory is a trefoil knot while in case of an irrational number, there is no closed solution.