What's so special about sine? (Concerning $y'' = -y$)
In an attempt to actually grok sine, I came across the $y''= -y$ definition.
This is incredibly cool, but it leads me to a whole new series of questions. Sine seems pretty prevalent everywhere in life (springs, sound, circles...) and I have to wonder, what's so special about the second derivative in this scenario?
In other words, why does nature / math seem to care more about the scenario where $y'' = -y$ instead of, say, $y' = -y$ or $y''' = y$?
Why is acceleration equal to negative the magnitude such a recurring theme in math and nature, while velocity equal to negative the magnitude ($y'=-y$) or jerk equal to negative the magnitude ($y'''=-y$) are seemingly unimportant?
In other words, what makes sine so special?
Note that this question also sort of applies to $e$, which satisfies $y'' = y$.
(Edit: Yes, I understand that $e$ and $\sin$ are closely related. I'm not looking for a relationship between $e$ and $\sin$.
Rather, I'm wondering why these functions in particular, which both arise from a relationship between a function and its own second derivative, are so prevalent. For example, do functions satisfying $y'''=-y$ also recur frequently, and I just haven't noticed them? Or is the second derivative in some way 'important'?)
This answer is perhaps more suited to physics.SE, but anyways.
I can see why you're wondering about the "specialness" of y'' = -y (or more generally -ky). Let me try to give an explanation of why it shows up all over the place.
Imagine a ball sitting at the bottom of a round well (looking like a U), its equation of motion is $y = 0$, its height is constant and not changing (let's say it is 0). Imagine you disturb this ball very slightly, you raise it to a very small height $\epsilon$ along the wall of the well. There's a force that will act on it to return it to 0. This force is a function of $\epsilon$, $F(\epsilon)$.
We can expand $F(\epsilon) = a_0 + a_1 \epsilon + a_2 \epsilon^2 + ...$ Noting that F(0) = 0, we get $a_0 = 0$. Since $\epsilon$ is small, $\epsilon^2$ and all higher powers are just very small.. So we will ignore them. So we get $F(\epsilon) \approx - k \epsilon$ (the negative sign is there because the function pulls the ball down). We can conclude that $F(y) = -k y$, around the equilibrium.
We know that the force is proportional to accleration (at low speeds). Therefore, $m y'' = -k y$ and thus $y'' = -c y$. This means that any small motion around a stable equilibrium is approximately a sinusoid. Now a circular motion is just two oscillations (one on each axis).
This is all just a consequence of the fact that our universe seems to favor equations of the second degree, since the force is proportional to the acceleration (not to the velocity and not to the change of acceleration). Why is this the case? One can trace that to the action (Hamiltonian) of the physical system or even to the conservation laws (which follow from the invariants, for example, that the physics won't change if we shift everything up or down, or if we try the experiment at a later time). Still these are just features of our universe.
From a physical perspective, this comes from the prevalence of simple harmonic oscillators. A SHO is a system in which the force is linear in $\mathbf{x}$ and directed opposite to the displacement from equilibrium. If you put that into Newton's second law, you get
$$\mathbf{F}_\text{net} = -k\mathbf{x} = m\ddot{\mathbf{x}}$$
With a suitable change of variables, this is just your differential equation $y''=-y$.
It turns out that nearly every bounded system in physics can be modeled as a simple harmonic oscillator, at least as a leading approximation. This is related to the tendency of physical systems to seek out a stable equilibrium in some potential function $V(\mathbf{x})$. At a stable equilibrium position, the first derivative of $V(\mathbf{x})$ vanishes, and because only potential differences are physically meaningful, the actual value at equilibrium can be set to zero, so the leading nontrivial term in a Taylor series expansion of the potential is the quadratic term, $V(\mathbf{x}) \approx V''(\mathbf{x}_0)(\mathbf{x}-\mathbf{x}_0)^2$. And because $\mathbf{F}=-\nabla V$, this potential gives you the behavior of a SHO for small displacements.
I guess this is not quite the expected answer, but it's definitely something very special about sine.
"What makes sine so special?" It is simply the wisest and the most moderate real function that could possibly exist !
Theorem
Let $f : \Bbb R \to \Bbb R$ a function of class $\mathcal C^\infty$.
Assume that :
- $f'(0) = 1$ ;
- for all non-negative integer $n$, the function $|f^{(n)}|$, the modulus of the $n$th derivative of $f$, is bounded by 1.
Then $f$ is the function sine. #
There are slightly stronger forms of this theorem, but I don't remember them... Everything is in the RMS, issue 116-3. Table of contents
In the following animation, the red arrow inside the circle traces out $\gamma(t)=(\cos(t),\sin(t))$ and red arrow outside the circle shows the tangent to the circle at $\gamma(t)$.
The blue arrow inside the circle shows $\gamma^{\,\prime}(t)$, which is parallel to the tangent at $\gamma(t)$. The blue arrow outside the circle shows the tangent to the circle at $\gamma^{\,\prime}(t)$.
The green arrow shows $\gamma^{\,\prime\prime}(t)$, which is parallel to the tangent at $\gamma^{\,\prime}(t)$.
$\hspace{4cm}$
Note that the green arrow is pointed in precisely the opposite direction as the red arrow inside the circle. Thus, we have $\gamma^{\,\prime\prime}(t)=-\gamma(t)$. Therefore, $\cos^{\,\prime\prime}(t)=-\cos(t)$ and $\sin^{\,\prime\prime}(t)=-\sin(t)$.
Periodic functions can be broken down into circular components. This is the main idea behind Fourier Analysis. If one of the circular components has a larger amplitude than the others, the whole function appears essentially circular. In two dimensions, if the amplitude in one of the directions is larger, the motion will appear one-dimensional, but it is still harmonic motion, where $x''=-kx$.
At least to me, this is what makes the circular functions ($\sin$ and $\cos$) so special; in a way, they are the basic periodic functions.