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:

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

${\frac {d}{dx}}\left[1\quad +\;\quad x\quad +\quad {\frac {x^{2}}{2\cdot 1}}\quad +\;\quad {\frac {x^{3}}{3\cdot 2\cdot 1}}\quad +\;\quad {\frac {x^{4}}{4\cdot 3\cdot 2\cdot 1}}\right]=$ $\;\;\quad \left[0\quad +\quad 1\quad +\quad 2{\frac {x^{1}}{2\cdot 1}}\quad +\quad 3{\frac {x^{2}}{3\cdot 2\cdot 1}}\quad +\quad 4{\frac {x^{3}}{4\cdot 3\cdot 2\cdot 1}}\right]=f(x)$ This function, $f(x)$ , is known to us as $\mathrm {e} ^{x}$ .

Sines and Cosines 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 $m$ 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 $x=-1$ 