Chapter 6.3, Problem 8E

Interpretation Introduction

Interpretation:

Find the acceleration and equilibrium position of the particle, and state the stability of the equilibrium position.

Concept Introduction:

The gravitational force between two objects is given as:

$\text{F =}\frac{{\text{G m}}_{\text{1}}{\text{m}}_{\text{2}}}{R^{\text{2}}}$

Here, ${\text{m}}_{\text{1}}{\text{ and m}}_{\text{2}}$ are masses of the objects, G is the universal gravitational constant, R is the distance between the objects, and F is the gravitational force.

If net force acting on it is zero, the system is said to be in equilibrium.

To check the stability of fixed point, use Jacobian matrix

$\left(\begin{array}{cc}\frac{\partial \dot{x}}{\partial x}& \frac{\partial \dot{x}}{\partial y}\\ \frac{\partial \dot{y}}{\partial x}& \frac{\partial \dot{y}}{\partial y}\end{array}\right)$

The point $(x^{*},y^{*})$ is said to be a stable fixed point if eigen values of Jacobian matrix evaluated at this point have negative real parts, and the point is said to be unstable if one of its eigenvalues has positive real part. If boththeeigenvalues are purely real, then the fixed point is saddle point. If the eigenvalues are purely imaginary, the fixed point has oscillating nature.