Complex Exponentials and Simple Harmonic Motion

The Superposition Principle

In lecture on Tuesday, we covered the use of complex exponentials to represent simple harmonic motion.  The reason we can use this notation is due to the following three facts:

1. The projection of 2 dimensional rotation at constant angular velocity onto one axis is identical to simple harmonic motion
2. the complex plane carries the same amount of information as the 2D plane
3. Euler’s formula provides a very concise packaging of this 2d motion into a single formula (just one expression rather than separate ones for the x and y component, for example)

This notation using the complex plane will provide useful for the following scenarios (as examples):

1. Superposition of multiple simple harmonic oscillators (avoiding trig)
2. Studying the behavior of 1D systems that include damping/dissipation and/or driving forces (rather inconvenient using trig) (Note that this is also the case in engineering when you have RLC circuits).
3. it is the fundamental tool in quantum mechanics, where the complex phases then have real physical meaning, so you better get used to it!!  😉

Here is a link to the mathematica file we are using this week:

https://jhubisz.expressions.syr.edu/phy360/wp-content/uploads/sites/5/2016/09/Week-2.nb

As we discussed briefly in class on Tuesday, in response to a question from Emily Syracuse, there is a very interesting and reason for why we can use this complex notation. Recall that in introducing complex notation we bring in extra information, taking 1D motion and turning it into 2D circular motion in the complex plane (the real and imaginary axes are the 2 dimensions).  In the end, we can always throw this extra information (the imaginary part) away to get at the physics (the projection of the circular motion onto one axis).

The reason for why this works is due to the fact that the equation of motion for a simple harmonic oscillator is linear.

Recall the simple harmonic oscillator equation is

There are no terms like or , or other higher powers of and its derivatives.

The SHO equation is linear because all potential energies near equilibrium are quadratic in displacement, meaning the force

is linear.

Equations like this – linear equations – obey a very important principle that is called the Superposition Principle.

The Superposition Principle states that if we have two solutions to an equation, the sum of the solutions is also a solution (or the difference, or any arbitrary linear combination of the two solutions).  You may be familiar with this principle from electromagnetism, and there is it also a result of the equations (Maxwell’s equations, in this case) being linear.

It is worth revisiting the “big picture” at this point:

• All systems with one degree of freedom near equilibrium have a potential energy which is quadratic in the displacement from equilibrium
• Thus the force is linear, and application Newton’s 2nd law yields a linear equation of motion
• Thus if and are both solutions, then so is or where and are any number at all (including imaginary numbers).

As we will see, it is very convenient to write solutions to the equation of motion for the SHO in the complex plane, even if the physical situation at hand refers only to real quantities, and so long as both the real and the imaginary part are both solutions, then this will work.  Circular motion in the complex plane is just the sum of two solutions, one is , the other is , and the coefficients and are and respectively.

An important consequence of the superposition principle is that of interference.  Recall now the demonstration where we look at what happens when 2 circular water waves are generated at 2 points some distance from each other.  When the crests coincide, you get a larger crest.  When a crest lies on top of a trough, you get no motion at all.  That is, the net effect of the two waves is the sum of the effects of each one individually.