# 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,

${\displaystyle e^{i\pi }=-1}$

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:

${\displaystyle y(t)=A\sin \omega t}$

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

The formula for exponential growth of a bank account with an initial principle ${\displaystyle P_{0}}$ and a growth rate of ${\displaystyle \alpha }$ is:

${\displaystyle P(t)=P_{0}e^{\alpha t}}$

WLOG we set ${\displaystyle \alpha =P_{0}=1}$ and find:

${\displaystyle \{\{f(x):f^{\,\prime }(x)=f(x){\text{ and }}f(0)=1\}\}}$

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

${\displaystyle f(x)=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2\cdot 1}}+{\frac {x^{3}}{3\cdot 2\cdot 1}}+{\frac {x^{4}}{4\cdot 3\cdot 2\cdot 1}}+...}$

Note that

${\displaystyle {\frac {d}{dx}}\left[1\quad +\;\quad x\quad +\quad {\frac {x^{2}}{2\cdot 1}}\quad +\;\quad {\frac {x^{3}}{3\cdot 2\cdot 1}}\quad +\;\quad {\frac {x^{4}}{4\cdot 3\cdot 2\cdot 1}}\right]=}$
${\displaystyle \;\;\quad \left[0\quad +\quad 1\quad +\quad 2{\frac {x^{1}}{2\cdot 1}}\quad +\quad 3{\frac {x^{2}}{3\cdot 2\cdot 1}}\quad +\quad 4{\frac {x^{3}}{4\cdot 3\cdot 2\cdot 1}}\right]=f(x)}$

This function, ${\displaystyle f(x)}$, is known to us as ${\displaystyle \mathrm {e} ^{x}}$.

## Sines and Cosines

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 ${\displaystyle m}$ on spring with spring constant ${\displaystyle k}$ is a sine or cosine wave:

${\displaystyle F_{y}=ma_{y}=m{\tfrac {d^{2}}{dt^{2}}}y=-ky}$

Again (WLOG) we set ${\displaystyle m=k=1}$ and replace the variable ${\displaystyle t}$ by ${\displaystyle x}$. Also, replace the position ${\displaystyle y}$ by the function ${\displaystyle g}$ or ${\displaystyle h}$, depending on whether we use a sine or cosine to describe the function.

Note that at ${\displaystyle x=0}$, ${\displaystyle sin(x=0)}$ equals zero, and the slope of ${\displaystyle cos(x)}$ also equals zero at ${\displaystyle x=0}$. This leads us on a search for infinite series solutions for two functions ${\displaystyle g(x)}$ and ${\displaystyle h(x)}$ that satisfy:

${\displaystyle \{\{g(x):g^{\,\prime \prime }(x)=-g(x){\text{ and }}g(0)=0\}\}}$
${\displaystyle \{\{h(x):h^{\,\prime \prime }(x)=-h(x){\text{ and }}h(0)=1\}\}}$

It is easy to show that:

${\displaystyle g(x)=x-{\frac {x^{3}}{3\cdot 2\cdot 1}}+{\frac {x^{5}}{5\cdot 4\cdot 3\cdot 2\cdot 1}}+...=\sin x}$
${\displaystyle h(x)=1-{\frac {x^{2}}{2\cdot 1}}+{\frac {x^{4}}{4\cdot 3\cdot 2\cdot 1}}+...=\cos x}$

## Completing the "proof" that ${\displaystyle e^{ix}=\cos x+i\sin x}$

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

Note what happens when we set ${\displaystyle x=-1}$