Question

Consider two identical oscillators, each with spring constant \(k\) and mass \(m\), in simple harmonic motion. One oscillator is started with initial conditions \(x_{0}\) and \(v_{j}\) the other starts with slightly different conditions, \(x_{0}+\delta x\) and \(v_{0}+\delta v_{1}\) a) Find the difference in the oscillators' positions, \(x_{1}(t)-x_{2}(t)\) for all t. b) This difference is bounded; that is, there exists a constant \(C\) independent of time, for which \(\left|x_{1}(t)-x_{2}(t)\right| \leq C\) holds for all \(t\). Find an expression for \(C\). What is the best bound, that is, the smallest value of \(C\) that works? (Note: An important characteristic of chaotic systems is exponential sensitivity to initial conditions; the difference in position of two such systems with slightly different initial conditions grows exponentially with time. You have just shown that an oscillator in simple harmonic motion is not a chaotic system.)

Short answer

Answer: The best bound for the absolute difference in positions of the two oscillators is 2A, where A is the amplitude of the oscillation.

Step-by-step solution

Find the general position functions of the oscillators

In simple harmonic motion, the position function is given by: $$ x(t) = A \cos(\omega t + \phi) $$ where \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(\phi\) is the phase shift. The angular frequency is given by: $$ \omega = \sqrt{\frac{k}{m}} $$ For the first oscillator with initial conditions \(x_0\) and \(v_0\), we can find \(A\) and \(\phi\) as follows: $$ x_0 = A \cos(\phi) $$ and $$ v_0 = -A\omega \sin(\phi) $$ For the second oscillator with initial conditions \(x_0 + \delta x\) and \(v_0 + \delta v_1\), we will have a similar set of equations but with a slight difference: $$ x_0 + \delta x = A \cos(\phi + \delta \phi) $$ and $$ v_0 + \delta v_1 = -A\omega \sin(\phi + \delta \phi) $$ where \(\delta \phi\) accounts for the difference in phase shift between the two oscillators.

Find the difference in positions of the oscillators

Now that we have the general position functions for both oscillators, we can find the difference between them: $$ x_1(t) - x_2(t) = A[\cos(\omega t + \phi) - \cos(\omega t + \phi + \delta \phi)] $$ Applying the sum-to-product formula: $$ x_1(t) - x_2(t) = 2A \sin\left(\frac{\delta \phi}{2}\right) \sin\left(\omega t + \phi + \frac{\delta \phi}{2}\right) $$

Find the expression for the constant C and the best bound

We want to find a constant \(C\) such that the absolute difference in positions is bounded by it for all time, t: $$ \left|x_1(t)-x_2(t)\right| \leq C $$ From the difference in positions found in the previous step, we have: $$ \left|2A \sin\left(\frac{\delta \phi}{2}\right) \sin\left(\omega t + \phi + \frac{\delta \phi}{2}\right)\right| \leq C $$ Since the maximum value of the sine function is 1, we can say that the best bound is when the two sine functions are at their maximum, which corresponds to the amplitude, \(2A\): $$ C = 2A $$ Thus, the best bound for the absolute difference in positions of the two oscillators is \(2A\), where \(A\) is the amplitude of the oscillation. This result shows that the simple harmonic oscillator is not a chaotic system since the difference in their positions does not grow exponentially with time but is bounded by a constant value.

Key concepts

Understanding the Oscillator in Physics

An oscillator in physics is a system that experiences repeated cycles or oscillatory motion, typically about an equilibrium position. This term encompasses a broad range of physical systems, from pendulums and springs to electronic circuits.

In the context of mechanics, a classical example of an oscillator is a mass attached to a spring, which moves back and forth when displaced from its equilibrium position. This type of motion is known as simple harmonic motion (SHM), characterized by its sinusoidal nature and constant frequency, regardless of amplitude.

An essential feature of an oscillator is its ability to store and transfer energy between different forms. For a mass-spring system, energy alternates between kinetic energy of the moving mass and potential energy stored in the spring's compression or extension. The interplay of these energy forms ensures that the motion continues until external forces, like friction, dampen the oscillations, leading to eventual cessation.Through understanding oscillators, students can grasp foundational concepts across various disciplines including mechanics, acoustics, and electrical engineering.

Demystifying Angular Frequency

The term angular frequency refers to the rate at which an object travels through its cycle in oscillatory motion, such as simple harmonic motion, and is denoted by the Greek letter omega (\f\(\fomega\f\)).In the equation \f\(x(t) = A \times \fcos(\fomega t + \fphi)\f\), the angular frequency \f\(\fomega\f\) is connected to the physical properties of the oscillator by the formula \f\(\fomega = \fsqrt{\frac{k}{m}}\f\), where \f\(k\f\) is the spring constant and \f\(m\f\) is the mass of the object.For students, it's useful to recognize that angular frequency is expressed in radians per second and is directly linked to the more familiar concept of frequency (\f\(f\f\)), which counts the number of cycles per second (Hertz). The two are related by the equation \f\(\fomega = 2\fpi f\f\), which bridges the gap between linear and angular measurements in oscillatory systems.By grasping angular frequency, students can predict how quickly an oscillator completes its cycle and delve deeper into understanding wave phenomena, resonance, and even quantum mechanics.

The Role of Phase Shift in Oscillatory Motion

The phase shift, denoted as \f\(\fphi\f\), is an essential concept in understanding oscillatory motion, particularly in determining the starting point or initial angle of the sine wave representing the motion.In the context of our problem with two oscillators, the phase shift affects the initial conditions of an object's oscillation, causing a 'shift' in the position of the waveform along the time axis. For example, when two oscillators with slightly different phase shifts (\f\(\fphi\f\) and \f\(\fphi + \f\fdelta\fphi\f\)) are compared, their motion is out of sync. This misalignment is particularly noticeable when drawing a comparison between their positions over time, as depicted by their sinusoidal waveforms.A clear understanding of phase shift is crucial for comprehending phenomena such as interference patterns in waves, the synchronization of multiple oscillators (like in clocks or musical instruments), and the modulation of signals in communications. For students tackling problems involving oscillatory motion, recognizing the impact of phase shift can be the key to solving complex problems involving wave superposition or resonance.