Question

Sketch the phase curves of the systems with potential energies \(U(x)=\) \(\pm x \sin x, U(x)=\pm \frac{\sin x}{x}, U(x)=\pm \sin x^{2}\)

Short answer

The main difference between the phase curves of the given systems with potential energies lies in the behavior of the conservative force that is derived from the potential energy functions. The first system (\(F(x) = x\cos x - \sin x\)) has oscillatory behavior influenced by a linear component, the second system (\(F(x) = \frac{x\cos x - \sin x}{x^2}\)) has oscillatory behavior that diminishes over time resembling a damped oscillation, and the third system (\(F(x) = -2x\cos x^2\)) has oscillatory behavior with increasing amplitude as x increases. The distinct conservative forces result in different phase curve behaviors for the systems.

Step-by-step solution

Calculate the conservative force from the potential energy functions

The conservative force F(x) can be obtained by calculating the negative derivative of the potential energy functions U(x) with respect to x. In this step, we will find the conservative force for each given potential energy function and then sketch the phase curves. 1. Potential energy function: \(U(x) = \pm x\sin x\) We differentiate it with respect to x and multiply by -1 to find the conservative force: \(F(x) = -\frac{d}{dx}(x\sin x) = x\cos x - \sin x\) 2. Potential energy function: \(U(x) = \pm \frac{\sin x}{x}\) We differentiate it with respect to x and multiply by -1 to find the conservative force: \(F(x) = -\frac{d}{dx}(\frac{\sin x}{x}) = \frac{x\cos x - \sin x}{x^2}\) 3. Potential energy function: \(U(x) = \pm \sin x^2\) We differentiate it with respect to x and multiply by -1 to find the conservative force: \(F(x) = -\frac{d}{dx}(\sin x^2) = -2x\cos x^2\)

Sketch the phase curves based on the conservative force

We will now utilize the conservative force from the potential energy functions to sketch the phase curves. The phase curves give an indication of how the system's state evolves over time when subjected to that force. 1. \(F(x) = x\cos x - \sin x\): This conservative force has an oscillatory component and a linear component. As x increases, both components play roles in determining the phase portrait. The phase curves will have oscillatory behavior like the sine and cosine functions but will also be influenced by the linear component x. 2. \(F(x) = \frac{x\cos x - \sin x}{x^2}\): Dividing by \(x^2\) causes the amplitude of the oscillatory behavior to decrease as x increases. As a consequence, the phase curves will show oscillatory behavior diminishing over time, resembling a damped oscillation. 3. \(F(x) = -2x\cos x^2\): Here, the potential energy function involves a sine function raised to the power of x^2. The conservative force has an oscillatory component and a linear component that is multiplied by -2x. The phase curves will have oscillatory behavior similar to that of \(F(x) = x\cos x - \sin x\) but with increasing amplitude as x increases due to the x^2 term inside the cosine function. Note that exact phase curves cannot be sketched in this format, and numerical methods should be used for more accurate depiction of the phase portraits.

Key concepts

Phase Curves

Phase curves are graphical representations that illustrate how the state of a dynamical system evolves over time. They are commonly used in the study of differential equations to visualize solutions. These curves are drawn in a phase space, which is a coordinate system where each point uniquely represents a state of the system.

For example, in a simple mechanical system, such as a swinging pendulum, the phase space might be defined by the pendulum's angle and its angular velocity. As the pendulum swings, its state traces out a continuous curve in this space.

• Phase space: A coordinate system representing all possible states of a system.
• State: A point in phase space showing the system’s current condition.
• Phase curve: Trajectory representing changes in the system’s state over time.

In our context, phase curves derived from potential energy functions describe how a system influenced by a conservative force (derived from potential energy) behaves over time. This understanding is crucial for predicting the future behavior of mechanical systems.

Potential Energy Functions

Potential energy functions play a central role in understanding the dynamics of physical systems. They represent the stored energy of the system due to its position or configuration, and are crucial for calculating the force acting on the system.

For the functions provided in the exercise, they each have a unique mathematical form that defines the potential energy landscape:

• For \(U(x) = \pm x \sin x\), the potential energy varies sinusoidally with x and is also proportionally scaled by x, creating a complex energy landscape.
• For \(U(x) = \pm \frac{\sin x}{x}\), the potential involves a diminishing amplitude for larger values of x due to the division by x, leading to different dynamics compared to the first function.
• For \(U(x) = \pm \sin x^2\), the potential is derived from the sine of squared x, introducing more rapid fluctuations in energy as x becomes larger.

The shape of these functions determines the conservative forces and the resulting phase curves.

Conservative Force

Conservative forces are special kinds of forces derived from potential energies. They are path-independent, meaning the work done by these forces depends only on the initial and final states, not the path taken.

In this exercise, we derived several conservative forces from given potential energy functions by taking the negative derivative with respect to position (x):

• From \(U(x) = \pm x \sin x\):The conservative force \( F(x) = x\cos x - \sin x \) combines both oscillatory and linear components.
• From \(U(x) = \pm \frac{\sin x}{x}\):The force \( F(x) = \frac{x\cos x - \sin x}{x^2} \) shows damped oscillations as the division by \(x^2\) reduces amplitude with increase in x.
• From \(U(x) = \pm \sin x^2\):The force \( F(x) = -2x\cos x^2 \) includes increasing oscillations amplified by the linear scaling factor \(-2x\).

These conservative forces are important for sketching phase curves, as they determine the system's dynamics over time.

Mathematical Modeling

Mathematical modeling involves creating mathematical representations of real-world systems to predict or understand their behavior. In physics, differential equations and energy functions often serve as the cornerstone of these models.

The exercise shows how mathematical models use differential equations to connect potential energy with calculated forces, revealing the system's dynamics. This is achieved through the combination of:

• Potential energy functions, illustrating the energy landscape.
• Conservative forces, calculated to understand how energy gets converted into motion.
• Phase curves, offering a visual depiction of system evolution over time.

By modeling these components with math, we gain insight into how a system will act under various conditions. These models are crucial for engineers and physicists when predicting behavior in real-world applications, such as designing safe structures or understanding natural phenomena.