Units and $ax^2L^2+bxL+c=0$ in the real world?

Question

It seems that most math equations that come from the real world usually come with dimensions, even though those dimensions are generally ignored. I'm speaking of general dimensions, which include not only time and space but also temperature, mass, volume, or anything else that can be measured. Given this, I've always wondered where polynomials come from, since they seem to violate dimensionality. For example if you are talking lengths you might have: $$ax^2L^2+bxL+c=0$$ where $L$ stands for a length dimension applied to the factor immediately proceeding. This equation implies adding an $L^2$ to an $L$ and that to a unitless quantity, which is invalid. Now you could obviously fix it by adding appropriate dimensions to the constants; for example: $$aL^{-1} x^2L^2+bxL+cL=0L$$ But I can't think of any real-world application that would produce this kind of equation. Of course geometry naturally gives rise to squares and cubes, but they are isolated squares or cubes; squares aren't added to linear values and cubes aren't added to squares.

So my question is, where do polynomials come from? Are there features of the physical world that give rise to them?

Answer 1

The ur-example of a polynomial model for something, as mentioned in another answer, is motion under constant acceleration. The reason this comes about is from Newton's first law of motion, which says that an object's acceleration is proportional to the force acting on it. Since acceleration is the second derivative of displacement with respect to time, that means that the equation that governs motion is:

$$\ddot{x} = \frac{F}{m}$$

where $F$ might be a function of other factors, but in the simplest case can be treated as a constant (and it helps that if you're close to the Earth's surface it's usually a reasonable approximation for the acceleration due to gravity).

When you do the integration, all of the constants of integration will naturally pick up the appropriate dimensions so that everything is consistent - an operation of $\frac{d}{dt}$ carries a dimension of $T^{-1}$, and an operation of $\int dt$ carries a dimension of $T$, so everything proceeds smoothly between the equations for position, velocity and acceleration.

Where things do wind up looking weird dimension-wise, you'll usually find that there are constants with built-in dimensions that cancel things out. For example, if you have harmonic motion like $x(t) = A \cos (\omega t + \phi)$, the dimension of $A$ will be $L$, the dimension of $\omega$ will be $T^{-1}$, and $\phi$ will be dimensionless, so that you're never taking the cosine of 5 seconds, for example.

Answer 2

Usually, the coefficients carry units such that all of the units work out out in the end.

Taking the classic equation from physics of an object under constant accelration.

$x = x_0 + v_0t + \frac 12 a t^2$

$a$ has units in $\frac {m}{s^2},$ $v$ has units in $\frac {m}{s}$ and $x,x_0$ have units in $m.$

Answer 3

What's the surface area of an open top box whose bottom is a square of side length $x$ and whose height is some fixed $h \in \mathbb{R}^+$?

Answer, $S(x) = x^2 + 4hx$.

So, by fixing one of the measurements e.g. let $h = 1/4$, we can attend to your dimensionality concern; in this case, you'd have $S(x) = x^2 + x$.

Answer 4

Dimension compatibility in physics is just a way to check that equations have some kind of validity. It is similar to type-checking in programming languages, and indeed some languages feature what they call "units", i.e. the ability to use dimensions and have the compiler automatically check them. We project data onto some simplified space (the space of physical units) because the equation validity is easier to check in this space.
We can make the unit system more precise by adding units, e.g. distinguishing between inertia mass and gravitational mass. Or simplify the unit system by suppressing units, e.g. not distinguishing anymore space from time, taking the velocity of light to $1$ (dimensionless) - of course then we would be more error-prone.

Dimensions in physics represent some invariance properties of phenomena, limited to scaling. We can actually extend this idea, however strange it may look.
For example representing change of frames by matrices is a similar idea: your coordinate frame is equivalent to a "unit"; when changing from one frame to another one, coordinates are multiplied by the inverse of the matrix that transforms the frames.
Or, we can extend units to addition: e.g. total energy is the sum of potential energy plus kinetic energy, we could have separate units for potential energy (let's say $E_p$) and kinetic energy (let's say $E_k$), and state that the unit of total energy is $E_p + E_k$. That would prevent people from forgetting one of the two terms. Why this is not used in practice, is probably because in many cases it would be a pain to distinguish between various kinds of energy.

Check Wikipedia's article on dimensional analysis for some ideas of possible extensions.
The classical use of units is described by the so-called theorem $\pi$.

Going back to polynomials, you can actually have polynomials in physics, as long as the variable is a unitless quantity. This is the same as $\log$, $\exp$, $\sin$ etc.: when these functions are used in physics, they are used with a unitless quantity. For example in thermodynamics, or for radioactive decay.
However I agree with you that it seems rare to have a polynomial equation with at least 3 terms (e.g. $x^2$, $x$, and a constant, with $x$ being dimensionless) in physics: thinking for it for a few minutes, I could not find any. Damped harmonic oscillator could perhaps be a candidate.

EDIT: indeed, cf. https://beltoforion.de/en/harmonic_oscillator/ :
If $\lambda$ is the oscillation frequency, $m$ the oscillator mass, $\mu$ the friction coefficient, $k$ the spring characteristic, we get
$\lambda^2 + \frac {\mu} m \lambda + \frac k m = 0$.