Chapter 6.5, Problem 12E

Interpretation Introduction

Interpretation:

To show that for a given system, energy is conserved. To show that origin is not an isolated fixed point. To show that origin is not surrounded by closed orbits and to sketch the actual phase portrait.

Concept Introduction:

Energy is often called as conserved quantity or a constant of motion or a first integral.

System for which conserved quantity exists is called a conservative system.

If for a given system $\dot{x}=f\left(x\right)$, a real valued continuous function $E\left(x\right)$ is constant on trajectories, then it is said to be a conserved quantity. In other words, it means $\frac{dE}{dt}=0$.

Also, it is necessary that $E\left(x\right)$ is not constant at every open set.