Question

Consider a mass \(m\) confined to the \(x\) axis and subject to a force \(F_{x}=k x\) where \(k>0 .\) (a) Write down and sketch the potential energy \(U(x)\) and describe the possible motions of the mass. (Distinguish between the cases that \(E>0 \text { and } E<0 .)\) (b) Write down the Hamiltonian \(\mathcal{H}(x, p)\), and describe the possible phase-space orbits for the two cases \(E>0\) and \(E<0\). (Remember that the function \(\mathcal{H}(x, p)\) must equal the constant energy \(E .\) ) Explain your answers to part (b) in terms of those to part (a).

Short answer

The potential energy is \( U(x) = \frac{k}{2}x^2 \) and motion is oscillatory for \( E > 0 \). The Hamiltonian yields elliptical phase-space orbits for \( E > 0 \); no orbits exist for \( E < 0 \).

Step-by-step solution

Determine the Potential Energy Function

To determine the potential energy function, we integrate the force with respect to position, using the relation that the force is the negative gradient of the potential energy: \( F_x = -\frac{dU}{dx} \). Given \( F_x = kx \), we have \( -kx = -\frac{dU}{dx} \). Integrating both sides gives \( U(x) = \frac{k}{2}x^2 \). This represents a parabolic potential centered at the origin.

Sketch the Potential Energy and Describe Motion

Sketch the potential energy \( U(x) = \frac{k}{2}x^2 \): It is a parabola opening upwards with its vertex at the origin. If the energy \( E > 0 \), the mass can move in bounded oscillations around the origin since the energy is insufficient to escape the potential well. If \( E < 0 \), which is physically unrealistic in classical mechanics for this scenario, the particle would not have enough energy to exist at any \( x \).

Formulate the Hamiltonian

The Hamiltonian \( \mathcal{H}(x, p) \) represents the total energy of the system and is given by \( \mathcal{H}(x, p) = \frac{p^2}{2m} + U(x) \). Substituting \( U(x) = \frac{k}{2}x^2 \), we obtain \( \mathcal{H}(x, p) = \frac{p^2}{2m} + \frac{k}{2}x^2 \). This Hamiltonian describes simple harmonic motion.

Analyze Phase-Space Orbits

Phase-space is a plot of \( x \) versus \( p \). For \( E > 0 \), the equation \( \mathcal{H}(x, p) = E \) describes an ellipse centered at the origin due to the form \( \frac{p^2}{2m} + \frac{k}{2}x^2 = E \). The orbits are closed and elliptical, indicating periodic motion. For \( E < 0 \), there are no possible phase-space orbits, as negative energy is not viable for real solutions.

Explain Phase-Space in Terms of Potential Energy

The elliptical phase-space orbits for \( E > 0 \) correspond to oscillations in potential energy described by \( U(x) = \frac{k}{2}x^2 \). These oscillations are confined within the potential well, representing motion that reflects back and forth in the potential. This matches the parabolic potential's confining nature. No phase-space motion exists for \( E < 0 \) since it's energetically impossible.

Key concepts

Potential Energy

Potential energy is a key concept in understanding harmonic oscillators. In this scenario, you have a force acting on a mass, which is described by the equation:

• \( F_x = kx \)

To find the potential energy \( U(x) \), you integrate the force with respect to position. This is because the force is the negative gradient of the potential energy. Using the given force, the integration appears as:

• \( -kx = -\frac{dU}{dx} \)

Upon integrating, the potential energy function becomes:

• \( U(x) = \frac{k}{2}x^2 \)

This function forms a parabolic shape, indicating that the potential energy increases as you move away from the origin. This parabola represents a potential well, where a mass can oscillate back and forth around the center.
For energy \( E > 0 \), the mass has enough energy for oscillations around the origin without escaping. Conversely, a scenario with \( E < 0 \) would be impossible in classical terms, as the energy would not allow the mass to exist within this potential.

Hamiltonian Mechanics

Hamiltonian mechanics is a formalism in physics that allows you to describe the total energy of a system. The Hamiltonian \( \mathcal{H}(x, p) \) for a simple harmonic oscillator combines both kinetic and potential energies as:

• \( \mathcal{H}(x, p) = \frac{p^2}{2m} + \frac{k}{2}x^2 \)

This formulation of the Hamiltonian establishes that the total energy is conserved within the system. The kinetic energy \( \frac{p^2}{2m} \) relates to the momentum \( p \), while the potential energy part \( \frac{k}{2}x^2 \) is derived from the force acting on the mass.
When you analyze this, for energies \( E > 0 \), it characterizes a system in simple harmonic motion. Such a system oscillates around its equilibrium point, and the Hamiltonian equation reflects this stable and periodic nature. It's important in this context because it allows the dynamics of the system to be understood as movements through a defined energy landscape.

Phase Space Analysis

Phase space gives you a powerful way to visualize the motion of a system. It uses position \( x \) and momentum \( p \) as axes to plot the state of the system. For a harmonic oscillator, the Hamiltonian gives:

• \( \mathcal{H}(x, p) = E \)

Where \( E \) is the total energy. For a system with \( E > 0 \), the phase-space plot reveals elliptical orbits. These ellipses represent closed, periodic paths, indicating stable oscillations around an equilibrium point.
Such visualizations help you understand how the kinetic and potential energies exchange roles as the system evolves. The elliptical trajectories demonstrate how the motion regularly confers momentum into position and vice versa, looping around the plot predictably.
However, when \( E < 0 \), there are no phase-space orbits because such a system state is not physically possible. The idea that negative total energy could result in any real-world motion is conceptually incorrect within this framework.