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,


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

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:
 * $$y(t)=A\sin\omega t$$

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 $$P_0$$ and a growth rate of $$\alpha$$ is:
 * $$P(t)=P_0 e^{\alpha t}$$

WLOG we set $$\alpha=P_0=1$$ and find:
 * $$\{\{f(x):f^{\,\prime}(x)=f(x) \text{ and } f(0)=1\}\}$$

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


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

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

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

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

$$ F_y= ma_y = m \tfrac{d^2}{dt^2} y = -ky$$

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

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


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

It is easy to show that:


 * $$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$$


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

Completing the "proof" that $$e^{ix}=\cos x + i\sin x$$
The rest of this proof is well-known. See for example Proof of Euler's Formula.

Note what happens when we set $$x=-1$$