Question

Consider an undamped, undriven simple harmonic oscillator - a mass \(m\) on the end of a spring whose force constant is \(k\). (a) Write down the general solution \(x(t)\) for the position as a function of time \(t .\) Use this to sketch the state-space orbit, showing the motion of the point \([x(t), \dot{x}(t)]\) in the two dimensional state space with coordinates \((x, \dot{x}) .\) Explain the direction in which the orbit is traced as time advances. (b) Write down the total energy of the system and use conservation of energy to prove that the state-space orbit is an ellipse.

Short answer

The state-space orbit is an ellipse, traced counterclockwise, showing the harmonic oscillator's position and velocity.

Step-by-step solution

General Solution of Simple Harmonic Oscillator

The simple harmonic oscillator equation is given by \( m\ddot{x} + kx = 0 \). The solution to this differential equation, considering it's undamped and undriven, takes the form \[ x(t) = A \cos(\omega t + \phi) \] where \( \omega = \sqrt{\frac{k}{m}} \) is the angular frequency, \( A \) is the amplitude, and \( \phi \) is the phase constant.

State-Space Representation

The state-space representation involves tracking both position \( x(t) \) and velocity \( \dot{x}(t) \). The velocity is given by differentiating \( x(t) \): \[ \dot{x}(t) = -A\omega \sin(\omega t + \phi) \]. The state-space orbit is described by coordinates \( (x(t), \dot{x}(t)) \), resulting in a parametric equation of the ellipse \[ \left( \frac{x}{A} \right)^2 + \left( \frac{\dot{x}}{A\omega} \right)^2 = 1 \].

Direction of the Orbit

As time advances, the point \( [x(t), \dot{x}(t)] \) moves counterclockwise on the ellipse in the \((x, \dot{x})\) state-space. This is because \( x(t) \) starts at \( A \) when \( \omega t + \phi = 0 \), and \( \dot{x}(t) \) is zero; as \( t \) increases, \( x(t) \) decreases and \( \dot{x}(t) \) becomes negative, leading the orbit smoothly counterclockwise.

Total Energy of the System

The total energy \( E \) of the system is given by the sum of kinetic and potential energies: \[ E = \frac{1}{2} m \dot{x}^2 + \frac{1}{2} k x^2 \]. Since energy is conserved, \( E \) is constant over time.

Proving State-Space Orbit is an Ellipse

The total energy equation \( \frac{1}{2} m \dot{x}^2 + \frac{1}{2} k x^2 = E \) can be rewritten as \( \frac{k x^2}{2E} + \frac{m \dot{x}^2}{2E} = 1 \), which takes the standard form of an ellipse \( \frac{x^2}{(A)^2} + \frac{\dot{x}^2}{(A\omega)^2} = 1 \) in the \( (x, \dot{x}) \) state space, demonstrating that the motion traces an ellipse.

Key concepts

State-Space Representation

In physics, the state-space representation provides a framework to analyze dynamic systems, like the simple harmonic oscillator, using both position and velocity. Imagine tracking a mass on a spring. As it moves back and forth, you want to capture more than just where it is. Therefore, you also keep track of how fast it’s moving. This dual focus is what we call state-space. Here, each point in the state-space is defined by two coordinates: position, denoted by \(x(t)\), and velocity, denoted by \(\dot{x}(t)\).

To fully comprehend it, recall that the position \(x(t)\) of an undriven, undamped simple harmonic oscillator is given by \(x(t) = A \cos(\omega t + \phi)\). The velocity, on the other hand, is the derivative of the position: \(\dot{x}(t) = -A\omega \sin(\omega t + \phi)\). This rich representation allows us to see the system's entire behavior as a single point moving in the state-space plane, defined by \(x(t)\) and \(\dot{x}(t)\). This movement is not random; it follows a specific path, revealing a deeper insight into the dynamics of the oscillator.

Conservation of Energy

The concept of energy conservation is a cornerstone in physics, guaranteeing that energy remains constant in an isolated system. For our simple harmonic oscillator, this conservation manifests in a clear way.

The total energy \(E\) in the system is the sum of its kinetic and potential energy. Mathematically, it's expressed as:

• Kinetic energy: \(\frac{1}{2} m \dot{x}^2\)
• Potential energy: \(\frac{1}{2} k x^2\)

Thus, the total energy is \[E = \frac{1}{2} m \dot{x}^2 + \frac{1}{2} k x^2\].

Importantly, for an undamped motion, this energy is conserved and does not change over time. It implies that the motion can continue indefinitely without any loss, making this conservation principle crucial in demonstrating the nature of the system’s motion in state-space.

Ellipse in State-Space

When we plot the movement of the mass in state-space, an intriguing geometric shape emerges: an ellipse.

To understand why, consider how energy constraints explain this motion. From energy conservation, we have \[\frac{1}{2} m \dot{x}^2 + \frac{1}{2} k x^2 = E\].
When manipulated, this equation resembles the standard form of an ellipse: \[\frac{x^2}{(A)^2} + \frac{\dot{x}^2}{(A\omega)^2} = 1\].
This means that every point \((x(t), \dot{x}(t))\) the system reaches in state-space lies on this ellipse.

In simpler terms, the same total energy is maintaining the system's path within the boundaries of this ellipse, dictating not just the outline but also details of its internal trajectory. Therefore, studying the ellipse helps us unpack the natural rhythm and limits of the oscillator’s motion.

Undamped Motion

In the context of our simple harmonic oscillator, undamped motion refers to the ideal state where no resistive forces (like friction or air resistance) are present. This means that energy isn't lost over time, ensuring perpetual motion.

Without damping, the oscillator swing from one side to the other endlessly. This is evident in the way the point \((x(t), \dot{x}(t))\) traces the ellipse in state-space. As time progresses, the trajectory doesn't shrink or change shape; it remains consistent, moving counterclockwise given physical laws governing the system. This perpetual cycle is a firsthand showcase of the energy's ongoing conservation in an undamped system.

While in real-world scenarios, perfect undamped motion is impossible due to inherent resistances, studying it theoretically helps simplify and understand the core dynamics of oscillatory systems. Thus, recognizing the undamped nature in models assists in grasping the underlying mechanics without added complexities of real-world damping factors.