Problem 24 - Advanced Engineering Mathematics

Question

(a) \(\mathbf{V}=\frac{a+i b}{\bar{z}}=\left(\frac{a x-b y}{x^{2}+y^{2}}, \frac{b x+a y}{x^{2}+y^{2}}\right)\) and since \(\left(x^{\prime}(t), y^{\prime}(t)\right)=\mathbf{V},\) the path of the particle satisfies $$\frac{d x}{d t}=\frac{a x-b y}{x^{2}+y^{2}}, \quad \frac{d y}{d t}=\frac{b x+a y}{x^{2}+y^{2}}$$ (b) Switching to polar coordinates, \\[ \begin{array}{l} \frac{d r}{d t}=\frac{1}{r}\left(x \frac{d x}{d t}+y \frac{d y}{d t}\right)=\frac{1}{r}\left(\frac{a x^{2}-b x y}{r^{2}}+\frac{b x y+a y^{2}}{r^{2}}\right)=\frac{a}{r} \\ \frac{d \theta}{d t}=\frac{1}{r^{2}}\left(-y \frac{d x}{d t}+x \frac{d y}{d t}\right)=\frac{1}{r^{2}}\left(\frac{-a x y+b y^{2}}{r^{2}}+\frac{b x^{2}+a x y}{r^{2}}\right)=\frac{b}{r^{2}} \end{array} \\] Therefore \(\frac{d r}{d \theta}=\frac{a}{b} r\) and so \(r=c e^{a \theta / b}\) (c) \(\frac{d r}{d t}=\frac{a}{r}<0\) if and only if \(a<0,\) and in this case \(r\) is decreasing and the curve spirals inward. \(\frac{d \theta}{d t}=\frac{b}{r^{2}}<0\) if and only if \(b<0,\) and in this case \(\theta\) is decreasing and the curve is traversed clockwise.

Step-by-Step Solution

Answer

Inward clockwise spiral if both \(a < 0\) and \(b < 0\).

Step 1: Understanding the Particle's Velocity

Given the velocity vector \( \mathbf{V} = \frac{a+i b}{\bar{z}} = \left( \frac{a x-b y}{x^{2}+y^{2}}, \frac{b x+a y}{x^{2}+y^{2}} \right) \), it can be broken into its components in Cartesian coordinates. These components represent the rate of change of position \((x, y)\) with respect to time \(t\). This leads to the differential equations: \( \frac{d x}{d t} = \frac{a x - b y}{x^{2} + y^{2}} \) and \( \frac{d y}{d t} = \frac{b x + a y}{x^{2} + y^{2}} \).

Step 2: Switch to Polar Coordinates

To simplify analysis, switch from Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\), where \( r \) is the radius and \( \theta \) is the angle. Note that \( x = r \cos \theta \) and \( y = r \sin \theta \). The provided transformations show \( \frac{d r}{d t} = \frac{a}{r} \) and \( \frac{d \theta}{d t} = \frac{b}{r^{2}} \), describing how \( r \) and \( \theta \) change with time.

Step 3: Differential Relation Between Radius and Angle

Using the expressions for \( \frac{d r}{d t} \) and \( \frac{d \theta}{d t} \), derive the relation \( \frac{d r}{d \theta} = \frac{a}{b} r \). Solving this differential equation yields \( r = c e^{a \theta / b} \), where \( c \) is a constant of integration. This describes the spiral path of the particle.

Step 4: Assess Direction of Motion

Determine the conditions for the particle's path regarding whether \( r \) and \( \theta \) increase or decrease over time. If \( \frac{d r}{d t} = \frac{a}{r} < 0 \), then \( a < 0 \) and radius \( r \) decreases, meaning the path spirals inward. If \( \frac{d \theta}{d t} = \frac{b}{r^{2}} < 0 \), then \( b < 0 \) and the angle \( \theta \) decreases, indicating a clockwise spiral path.

Key Concepts

Phase Plane Analysis

Phase plane analysis is a valuable tool used to study the behavior of systems of differential equations. In the context of a differential equation, it involves examining the system's trajectory in a two-dimensional space represented by the variables of the system, often denoted as \((x, y)\).

• What is a Phase Plane? - The phase plane is a graphical representation where each point \((x, y)\) corresponds to a particular state of the system. It allows us to visualize and analyze the system's dynamic behavior over time.
• Visualizing System Dynamics - By plotting the phase plane, we can observe trajectories and determine stability, presence of equilibrium points, and limit cycles.
• Interpretation of Trajectories - In the phase plane, trajectories can show whether the system behavior tends towards stability (approaching an equilibrium point), instability (diverging away), or complex patterns like oscillations.

Phase plane analysis simplifies the complexity of solving differential equations by providing a visual interpretation. It helps us understand the long-term behavior of systems, which is essential in engineering and physics applications.

Polar Coordinates

Polar coordinates are an alternative way of defining a location in a plane using a radius and an angle. This system is particularly useful in problems involving circular or rotational symmetry.

• Why Use Polar Coordinates? - In situations where phenomena have radial symmetry, polar coordinates can simplify mathematical expressions and solutions.
• Basic Transformation - The transformation from Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\) is given by \(x = r \cos \theta\) and \(y = r \sin \theta\).
• Working with Polar Equations - In our example, the differential equations in polar coordinates show how radius \(r\) and angle \(\theta\) change over time, which can be more intuitive to analyze than Cartesian coordinates in certain contexts.

Switching to polar coordinates can turn complex motion described by Cartesian coordinates into simpler, often more manageable problems, allowing us to see patterns and solutions that are otherwise hard to spot.

Spiral Dynamics

Spiral dynamics describe the patterns of movement that look like a coil or spiral. Studying these patterns is crucial when analyzing systems that exhibit circular or oscillatory behavior.

• Understanding Spiral Patterns - Within a system, spiral dynamics refer to trajectories in phase space that resemble spirals. They represent changes in radius and angular position over time.
• Inward and Outward Spirals - A system can spiral inward, indicating a decrease in radius, or outward, indicating an increase. This is determined by the sign and relative magnitude of parameters in the governing equations.
• Clockwise vs. Counter-Clockwise - The direction of spiraling (clockwise or counter-clockwise) is influenced by the angular velocity term. If \(b < 0\), the spiral moves clockwise; if \(b > 0\), it moves counter-clockwise.

Spiral dynamics help us understand the comprehensive motion of dynamic systems, including oscillations and rotations. This can apply to real-world scenarios such as weather patterns, mechanical systems, and more.