Problem 26 - Advanced Engineering Mathematics

Question

Solve the given nonlinear plane autonomous system by changing to polar coordinates. Describe the geometric behavior of the solution that satisfies the given initial condition(s). \(x^{\prime}=y-\frac{x}{\sqrt{x^{2}+y^{2}}}\left(4-x^{2}-y^{2}\right)\) $$ \begin{aligned} &y^{\prime}=-x-\frac{x}{\sqrt{x^{2}+y^{2}}}\left(4-x^{2}-y^{2}\right) \\ &\mathbf{X}(0)=(1,0) ; \mathbf{X}(0)=(2,0) \end{aligned} $$

Step-by-Step Solution

Answer

The orbits are circles; solutions revolve clockwise with constant radii determined by their initial conditions.

Step 1: Convert to Polar Coordinates

In polar coordinates, \( (x, y) \) becomes \( (r \cos\theta, r \sin\theta) \). We substitute these into the system:- \( x = r \cos\theta \) and \( y = r \sin\theta \).Now express the derivatives:- \( x' = r' \cos\theta - r \sin\theta \theta' \)- \( y' = r' \sin\theta + r \cos\theta \theta' \) This leads to changes in the formula for \( x' \) and \( y' \). The key is to then express these in terms of \( r \) and \( \theta \).

Step 2: Express the System in Polar Form

Substitute the polar expressions of \( x \) and \( y \) in our original system:- \( x' = y - \frac{x}{\sqrt{x^2 + y^2}}(4 - x^2 - y^2) \)- \( y' = -x - \frac{x}{\sqrt{x^2 + y^2}}(4 - x^2 - y^2) \)Since \( \sqrt{x^2 + y^2} = r \), the system translates to:- \( r' = 0 \)- \( \theta' = -1 \)This means \( r \) is constant and only \( \theta \) varies linearly over time.

Step 3: Analyze Initial Conditions

Given the initial conditions \( \mathbf{X}(0) = (1,0) \) and \( \mathbf{X}(0) = (2,0) \), we convert to polar coordinates:- For \( (1,0) \), \( r = 1 \) and \( \theta = 0 \).- For \( (2,0) \), \( r = 2 \) and \( \theta = 0 \).Since \( r' = 0 \), the radius doesn't change, the behavior is a rotation with radius \( 1 \) and \( 2 \) respectively.

Step 4: Describe the Geometric Behavior

The solutions describe circles centered at the origin. For the initial condition \( (1,0) \), the solution is a circle of radius 1. Similarly, for \( (2,0) \), it is a circle of radius 2. Since \( \theta' = -1 \), the motion is clockwise.

Key Concepts

Polar Coordinates

Polar coordinates provide a way to represent points in a plane using a radius and an angle, which is particularly useful for circular or rotational scenarios. Instead of using the traditional Cartesian coordinates

• Cartesian: given as \( (x, y) \)
• Polar: expressed as \( (r \cos \theta, r \sin \theta) \)

You convert a point from the Cartesian system \( x \) and \( y \) to polar by using \( x = r \cos \theta \) and \( y = r \sin \theta \).
In this exercise, converting to polar coordinates simplifies our nonlinear system greatly. \( r \) represents the distance of the point from the origin, while \( \theta \) is the angle from the positive \( x \)-axis. This enables analysis in terms of radii and angles, revealing underlying patterns or symmetries.

Geometric Behavior of Solutions

When analyzing geometric behavior, the focus is on how the solutions evolve over time. The expressions \( r' = 0 \) and \( \theta' = -1 \) derived from polar transformations reveal key insights:

• \( r' = 0 \): The radius remains constant; points do not move closer to or further from the origin.
• \( \theta' = -1 \): The angular rate (or change in direction) is constant, indicating uniform rotation.

This behavior confirms that points travel in circles centered at the origin. Their size is dictated by the initial radius \( r \).
The evolution is predictable and regular, often translating complex nonlinear motions into simple circular paths.

Initial Conditions Analysis

Initial conditions are crucial as they set the starting point for determining the trajectory of solutions. Here,\( \mathbf{X}(0) = (1,0) \) and \( \mathbf{X}(0) = (2,0) \) call for an initial analysis.
For \( (1,0) \):

• Convert to polar: \( r = 1 \), \( \theta = 0 \)
• Implies a circular path with radius 1

For \( (2,0) \):

• Convert to polar: \( r = 2 \), \( \theta = 0 \)
• Implies a circular path with radius 2

These conditions show how initial radius \( r \) defines the size of the circle on which points rotate. Despite the point's starting angle, the radius decides the trajectory's scale.

Dynamical Systems Rotation

Rotational dynamics refer to the circular movement derived from the system's equations. In this case, the system is described by

• \( r' = 0 \) - no radial change
• \( \theta' = -1 \) - negative sign denotes clockwise rotation.

This simple setup converts the complexity of nonlinear systems into manageable dynamics.
Because \( \theta \) changes linearly with time, the circle traverses steadily. The direction is counterclockwise if \( \theta' > 0 \), otherwise clockwise.
These polar dynamics imply a complete understanding of movements typical in systems, ensuring clear visualizations and precise forecasts.