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

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

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]

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