Could somebody give me an intuitive physical interpretation of higher order derivatives (from 2 and so on), that is not related to position – velocity – acceleration – jerk – etc?
Kinematics – Physical Intuition for Higher Order Derivatives
accelerationdifferentiationjerkkinematicssoft-question
Related Solutions
I) Disclaimer: In this answer, we will only consider the mathematically idealized classical problem, which is interesting in itself, although admittedly academic and detached from actual physical systems.
II) It is in principle possible that all time derivatives of the position $x(t)$ vanishes at $t=0$, and yet the particle does move away (from where it was at $t=0$).
$\uparrow$ Fig. 1: A plot of position $x$ as a function of time $t$.
Example: Assume that the position is given by the following infinitely many times differentiable function
$$ x(t)~:=~\left\{ \begin{array}{ccl} \exp\left(-\frac{1}{|t|}\right)&\text{for} & t~\neq~ 0, \\ \\ 0 &\text{for} & t~=~ 0. \end{array} \right. $$
Note that the Taylor series for the $C^{\infty}$-function $x:\mathbb{R}\to\mathbb{R}$ in the point $t=0$ is identically equal to zero.$^1$ So the function $x$ and its Taylor series are different for $t\neq 0$. In particular, we conclude that the smooth function $x$ is not a real analytic function.
III) If you like this Phys.SE question you may also enjoy reading about What situations in classical physics are non-deterministic? , Non-uniqueness of solutions in Newtonian mechanics and Norton's dome and its equation.
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$^1$ Sketched proof that $x\in C^{\infty}(\mathbb{R})$: Firstly, obviously, $x$ is $C^{\infty}$ for $t\neq 0$. By induction in $n\in\mathbb{N}_0$, for $t\neq 0$, it is straightforward to deduce that the $n$'th derivative is of the form $$x^{(n)}(t)~=~\frac{P_n(t,|t|)}{Q_n(t,|t|)}x(t), \qquad t\neq 0,$$ for some polynomials $P_n$ and $Q_n\neq 0$. Secondly, it follows that the $(n\!+\!1)$'th derivative at the origin $$x^{(n+1)}(0)~=~\lim_{t\to 0} \frac{x^{(n)}(t)}{t}~=~0$$ exists and is zero "because exponentials beat polynomials". $\Box$
Best Answer
If you want a simple intuitive explanation, you can get a lot from vehicles.
In a car traveling at a constant speed, suppose there is a white dot painted on the top of the steering wheel. If that dot is in the center, you are traveling in a straight line. If you turn it some angle to the left, say 90 degrees, then the car is traveling in a circular arc at a constant lateral acceleration. That is the second derivative of lateral position.
If you turn the steering at a constant rate from 0 degrees to 90 degrees, then the rate of lateral acceleration is changing constantly while you are turning the wheel. That is jerk, and it is constant because you are turning the steering wheel at constant speed. It is the third derivative of lateral position.
(While you are doing this, the path traced by the car is a spiral of linearly increasing curvature. Highway and railway curves are built using these spirals to connect segments of constant curvature. Such a spiral gives a place to gradually bank the roadway, and without it drivers tend to cross lanes, and trains actually jerk when starting or ending a curve.)
However, if you don't turn the steering wheel at a constant rate, but rather accelerate it leftward from 0 degrees until you are turning it quickly at 45 degrees, and then decelerate it until it reaches 90 degrees, then you are giving it a doublet of snap, first positive, then negative, and that is the fourth derivative of lateral acceleration.
Another vehicle to illustrate it is a submarine having bow planes to control depth. Suppose there is a motor that rotates the bow planes at a constant speed, up, down, or 0. The angle of the bow planes determines the pitch rate of the submarine. The pitch angle of the submarine determines the rate of change of depth.
So the pitch angle is proportional to the first derivative of depth, the bow plane angle determines the second derivative, and the speed of the bow plane motor is the third derivative.
Also, take a rocket with gimballed engines. If the thrust line of a rocket engine does not go through the rocket's center of mass, then it produces angular acceleration, or the second derivative of directional orientation. There are motors that move the engine gimbals, and the rate at which they move them determines the third derivative of direction.
I'm sure you can think of other examples.