Chapter 7.2, Problem 9E

Interpretation Introduction

Interpretation:

For the system, decide whether it is gradient or not. If it is, find V and sketch the phase portrait. Also sketch the equipotential surfaces $\text{V=constant}$

Concept Introduction:

Gradient systems: If the system can be written in the form $\dot{\text{x}}=-\nabla \text{V}$ for continuously differentiable, single-valued scalar function $\text{V(x)}$, then the system is called as gradient system with a potential function V.

$\nabla V$ can be written as

$\nabla \text{V=}\frac{\delta V}{\delta \text{x}}+\frac{\delta V}{\delta \text{y}}$

$\dot{\text{x}}\text{= -}\frac{\delta V}{\delta \text{x}}$ and $\dot{y}\text{=}-\frac{\delta V}{\delta \text{y}}$

$\therefore \nabla \text{V=}-\dot{\text{x}}-\dot{\text{y}}$

$\text{V=}-{\displaystyle \int \dot{\text{x}}\text{ dx}}-{\displaystyle \int \dot{\text{y}}}dy$

For system to be gradient

$-\nabla \text{V= }\left(\text{-}\frac{\delta V}{\delta \text{x}}\text{,-}\frac{\delta V}{\delta \text{y}}\right)=\left(f\left(x,y\right),g\left(x,y\right)\right)$

$\text{-}\frac{\delta V}{\delta \text{x}}=\text{f}$ and $-\frac{\delta V}{\delta \text{y}}=g$

$\frac{\partial \text{ f}}{\partial \text{y}}=-\frac{{\partial }^2\text{V}}{\partial \text{y}\partial \text{x}}=\frac{\partial \dot{x}}{\partial y}$

$\frac{\partial g}{\partial x}=-\frac{{\partial }^{2}V}{\partial x\partial y}=\frac{\partial \dot{y}}{\partial x}$

If f and g both are smooth on ${ℝ}^2$ and $\frac{\partial \dot{x}}{\partial \text{y}}=\frac{\partial \dot{y}}{\partial x}$, then the system is said to be gradient.