All that is known in the entire field of mathematics. But yeah, mostly just mathematics. I think sorting algorithms is a solved problem, but you can say that's just math as well.

Even math systems are engineered somewhat: we can always invent a new and better math. It has happened a few times in history.

Well, Newton's laws of motion suffice for non-relativistic motion. Maxwell's equations, I think, suffice for typical electronic/magnetic fields. Judging from all the labs I did or TA'd back then, we have a pretty good handle on friction and inclined planes. While there's a lot of "science at the edges" that's ongoing, much of the everyday, classical things have a suitable solution.

As others have written, modern Engineering could not be done without the existence of such working solutions.