Hacker News Reader

Comment by gweinberg | Hacker News Reader

It's fortunate that we so seldom need to solve cubic or higher order polynomial equations. Quadratic equations come up in physics all the time, because it's very common to have a constant acceleration. But when does one ever come across a constant jerk?

View on HN

As others pointed out, this is just a fallacy. Physics (as well as Math, CS, and all other disciplines) is full of non-linear, chaotic problems that are way too hard to solve, so academia doesn't like to teach them to new students and even experts aren't delighted to deal with them, because it's frustrating that we can't solve them (in a "beautiful" and "elegant" way). My own hot take is that Physics is stuck on a theory of everything for the last 100 years, because we are stuck on some fundamental limitations what Math is able to model.

>My own hot take is that Physics is stuck on a theory of everything for the last 100 years, because we are stuck on some fundamental limitations what Math is able to model.

Your previous assumption reads more like math being able to model it, but scientists preferring elegant, aesthetic and simplistic solutions.

Looking at my PhD physics neighbor in student housing a few years back, it doesn't seem true though. He was juggling multi-page equations for his quantum field theory thesis.

In control engineering it's more true though. I remember one of my professors say that many people only use linear controls because there they can "prove" the stability of the system. In contrast to the nonlinear, i.e. everything else, control theory where such proofs were harder to get by.

However, he continued, proving the stability of a linear controller interacting with a linearized system could never "prove" the system's stability as a whole.

From experience linear controls is probably more popular because there's a larger set of tools in the toolbox, so to speak, and the nature of the systems is that this is (usually) close enough for modelling nonlinear systems which are close enough to linear (in fact you may introduce nonlinear terms in your controller to try to remove nonlinearities in your system). It's not just having a 'proof' of stability, it's having methods to analyse the system and optimise the controller, which are much more difficult to apply in general to non-linear systems. It's much like how engineers will generally just stick to classical models of physics despite them being 'wrong', because in practice they are close enough and also much easier to deal with.

> because we are stuck on some fundamental limitations what Math is able to model.

This seems more a limitation of what (human) mathematicians are able to model, rather than something fundamental in math itself. Perhaps in a few centuries we will have invented mathematical tools that allow the mind to grasp nonlinear systems better, similar to how the invention of better construction tooling has enabled the construction of ever larger buildings.

On the other hand, physics is filled with problems that are far worse behaved than even large polynomials. 3+ massive bodies interacting under Newtonian gravity. 2+ massive bodies interacting in General relativity [0], anything in quantum field theory, almost all differential equations you run into outside of a textbook on differential equations.

[0] To be far, I'm not actually positive that the 2 body problem in GR is actually computationally harder than massive polynomials, but there is no exact solution to it; and understanding the approximation is definitely harder.

I've worked with several.

I did audibly chuckle at this one. Good job, doesn't happen to me often online!

I think I waited 25 years for the opportunity to unleash this particular dad joke.

For when one would come across a constant jerk - how about the usage case of putting a stopped vehicle into motion? You gradually push the accelerator or throttle from zero to maximum at a constant pace - that's increasing the acceleration linearly (ignoring friction/drag), thus constant jerk, which the occupants will perceive as a smooth start. This probably doesn't often get solved numerically, but it's something you do instinctively do.

>It's fortunate that we so seldom need to solve cubic or higher order polynomial equations.

We do, it's just that we choose different formulations for problems which require such solutions, or straight up don't solve them.

Think about the unsolvable n-body problem from classical mechanics. An equivalent of that problem exists in most scientific disciplines. We just choose not to think of those problems from the polynomial solving angle.

I always wondered why that is for a long time (in basic physics that at least). It wasn't until grad school that I understood that position and velocity are enough to uniquely determine a trajectory on a manifold.

This is super interesting, is this generalizable to n-dimensional manifolds?

Yep! More specifically I'm basically talking about geodesics on Riemannian manifolds.

IMO the bigger part is that numerical root finding works really well in practice. There's plenty of times you need to solve for roots of equations, but getting within 1e-6 is often good enough.

A rocket could be an example, assuming constant thrust and linearly decreasing mass, the decreasing mass would result in constant jerk.

>when does one ever come across a constant jerk?

Read the news lately?

Doesn't every startup come bundled with one?

Reply on news.ycombinator.com