User:Guy vandegrift/2019/Euler's equation - physical explanation

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Euler's equation can be explained using physical arguments based on exponential growth and the simple harmonic oscillator.

Simple Harmonic Motion Orbit.gif

In essence, we aim to "prove" the remarkable formula,

We must use quotation marks around words like "proof" in this discussion because this will not be a mathematical proof. Instead, we use an observation. When a pendulum is set in motion, and a pencil is put on a rotating disk at the right speed and radius, it can be noted that the motion can be described as:

This is shown in the figure for a mass/spring system. When combined with F=ma, physics can be used to construct our "proof".

To get in the mood for this, visit Casimir effect (zeta-regularization) and it's link to the "Astounding" YouTube video on Wikiversity.

Exponential growth[edit]

The formula for exponential growth of a bank account with an initial principle and a growth rate of is:

WLOG we set and find:

To understand this notation, see Set notation. Next, we try this infinite series:

Note that

This function, , is known to us as .

Sines and Cosines[edit]

This image shows approximations to sin x where the series is terminated at 1, 3, 5, 7, 9, 11 and 13.

Let us begin with the assumption that the solution to Newton's second law for a mass on spring with spring constant is a sine or cosine wave:

Again (WLOG) we set and replace the variable by . Also, replace the position by the function or , depending on whether we use a sine or cosine to describe the function.

Note that at , equals zero, and the slope of also equals zero at . This leads us on a search for infinite series solutions for two functions and that satisfy:

It is easy to show that:

Completing the "proof" that [edit]

The rest of this proof is well-known. See for example Proof of Euler's Formula.

Note what happens when we set