Problem 3 - Classical Mechanics

Question

The relativistic equivalent of the simple harmonic oscillator equation for a spring with constant \(k\) and a rest mass \(m_{0}\) attached is $$ \frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{m_{0} y}{\sqrt{1-y^{2} / c^{2}}}\right)+k x=0 \quad \text { with } \quad \dot{x}=y $$ where \(c\) is the speed of light. Show that the phase trajectories are given by $$ m_{0} c^{2} / \sqrt{1-y^{2} / c^{2}}+\frac{1}{2} k x^{2}=\text { constant } $$ and sketch the phase portrait for this system.

Step-by-Step Solution

Answer

Q: Show that the phase trajectories for the given relativistic spring equation are given by the expression \(\frac{m_{0} c^{2}}{\sqrt{1 - y^{2} / c^{2}}} + \frac{1}{2} k x^{2} = const\). A: To show that the phase trajectories for the given relativistic spring equation match the required expression, we followed these steps: 1. Analyze the given equation: The equation is a relativistic spring system with force proportional to displacement, \(x\), and rest mass, \(m_{0}\). The term \(\dot{x}\) is the velocity of the mass and is equal to \(y\). 2. Substitute \(\dot{x} = y\): We eliminate \(\dot{x}\) from the equation using the substitution, \(\dot{x}=y\), and rewrite the equation as $$\frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{m_{0}\dot{x}}{\sqrt{1 - \dot{x}^{2} / c^{2}}}\right)+k x=0$$ 3. Integrate both sides over time: Integrating both sides of the equation with respect to time, \(t\), we get: $$\frac{m_{0} c^{2}}{\sqrt{1 - \dot{x}^{2} / c^{2}}} + \frac{1}{2} k x^{2} = const$$ 4. Replace \(\dot{x}\) with \(y\): Finally, we replace \(\dot{x}\) back to \(y\): $$\frac{m_{0} c^{2}}{\sqrt{1 - y^{2} / c^{2}}} + \frac{1}{2} k x^{2} = const$$ This expression represents the phase trajectories for the given relativistic spring equation.

Step 1: Analyze the given equation

The given equation is: $$ \frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{m_{0} y}{\sqrt{1-y^{2} / c^{2}}}\right)+k x=0 \quad \text { with } \quad \dot{x}=y $$ This equation represents a relativistic spring system with force proportional to displacement, \(x\), and a rest mass, \(m_{0}\). The term \(\dot{x}\) is the velocity of the mass, and it is equal to \(y\).

Step 2: Substitute \(\dot{x} = y\)

We can eliminate \(\dot{x}\) from the equation using the substitution, \(\dot{x}=y\), and rewrite the given equation as follows: $$ \frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{m_{0}\dot{x}}{\sqrt{1 - \dot{x}^{2} / c^{2}}}\right)+k x=0 $$

Step 3: Integrate both sides over time

Now, we integrate both sides of the equation with respect to time, \(t\), to obtain: $$ \int \frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{m_{0}\dot{x}}{\sqrt{1 - \dot{x}^{2} / c^{2}}}\right) dt + \int k x dt = \int 0 dt $$ Evaluating the integrals yields: $$ \frac{m_{0} c^{2}}{\sqrt{1 - \dot{x}^{2} / c^{2}}} + \frac{1}{2} k x^{2} = const $$

Step 4: Replace \(\dot{x}\) with \(y\)

Replace \(\dot{x}\) back to \(y\): $$ \frac{m_{0} c^{2}}{\sqrt{1 - y^{2} / c^{2}}} + \frac{1}{2} k x^{2} = const $$ This expression represents the phase trajectories for the given relativistic spring system.

Step 5: Sketch the phase portrait

The phase portrait is a plot of \(x\) versus \(y\), which represents the state of the system with respect to position and velocity. Since the phase trajectories are composed of constant energy levels, in this case, the phase portrait will display a set of concentric ellipses. The ellipses will be centered at the origin (0,0) and will have their semi-major axes in the \(x\) direction, representing the position. The \(y\) direction represents the velocity of the mass, with higher values corresponding to higher velocities. The phase portrait provides insight into the dynamic behavior of the spring system, where each ellipse represents constant energy and the clockwise motion of the mass within each ellipse reflects the oscillatory dynamics of the harmonic oscillator.

Key Concepts

Phase Trajectories

In studying the movement of a physical system like the relativistic harmonic oscillator, phase trajectories are a crucial concept. These represent the path a system's state follows over time in phase space, which is a graphical representation of all possible states of a system. For the oscillator in question, the trajectory is defined by the coupled oscillator equation and its relativistic correction factor due to high velocities approaching the speed of light.

The resulting trajectory equation is given by:
\[\begin{equation}\frac{m_{0} c^{2}}{\sqrt{1 - y^{2} / c^{2}}} + \frac{1}{2} k x^{2} = \text{constant}\end{equation}\]This equation encapsulates an energy balance involving kinetic energy, encoded in the velocity term \( y \), and potential energy, represented by the displacement \( x \). The 'constant' reflects the conservation of energy within the system. In simpler terms, as the mass oscillates, its speed and position change, but the total energy remains the same. Thus, phase trajectories help visualize energy conservation in dynamic systems.

Special Relativity

The pivotal principle of special relativity, introduced by Albert Einstein, tells us that the laws of physics are the same for all non-accelerating observers and that the speed of light in a vacuum is the same no matter the speed at which an observer travels. This theory has profound implications for systems moving at high velocities, particularly close to the speed of light (\( c \)).

The given equation for the relativistic oscillator includes a factor of \( \sqrt{1-y^{2}/c^{2}} \) which corrects for relativistic effects, manifesting as the velocity (\( y \)) of the oscillating mass approaches closer to \( c \). In this regime, the mass experiences time dilation and length contraction, assuming that it is moving at a significant fraction of \( c \). These corrections are what distinguish the relativistic harmonic oscillator from its classical counterpart and affirm that special relativity must be accounted for in high-speed environments.

Simple Harmonic Motion

At the heart of oscillatory systems like pendulums and springs lies simple harmonic motion (SHM). It is characterized by the restorative force being proportional to the displacement but acting in the opposite direction, which can be mathematically expressed for a spring system as Hooke's Law:\( F = -kx \), with \( k \) being the spring constant and \( x \) the displacement from equilibrium.

In a classical, non-relativistic context, the mass on a spring exhibiting SHM would follow a sinusoidal pattern for both its position and velocity over time. Its phase trajectories, on the other hand, would be simple circles or ellipses in phase space, illustrating a predictable and repetitive path. However, in the relativistic case, the mass's increased speed demanding a relativistic treatment adds complexity, giving rise to the modified equation provided and a resulting phase trajectory that deviates from the classical ellipse.

Phase Portrait

A phase portrait serves as a snapshot encapsulating all possible states of a dynamical system. It's particularly insightful for understanding how a system evolves over time. In the context of the relativistic harmonic oscillator, the phase portrait would illustrate all conceivable positions (\( x \)) and momenta (\( y \)) of the attached mass.

Drawing the phase portrait involves plotting \( x \) against \( y \), where each point on the plot represents a specific state of the system. For the relativistic harmonic oscillator, we would see a family of similar curves, each corresponding to a different energy level, with the shape of these curves reflecting the relativistic nature of the motion—an adaptation of the ellipses found in classical SHM but with relativistic corrections. Understanding this visual representation allows students to better grasp the dynamics of the oscillator beyond what pure equations can convey.