Your second point is very interesting! Can you point me to any problems of that type? I’d be interested in learning how to make a mathematical model of something, and then simplify parts of it with various assumptions.
The one I can think of offhand is the pendulum problem, where sin(theta) is approximately equal to theta for small values. But you made it sound like there are problems with multiple parts, and many different simplifications.
It's a set of smart physical observations/assumptions that allow you to find closed form solutions for the Navier Stokes equations in fluid mechanics https://en.wikipedia.org/wiki/Navier–Stokes_equations for the important case of flow close to some body, such as an airfoil.
There definitely are. The simple pendulum (or rather harmonic oscillators) are the problem we want to transform harder stuff to, because we know how to solve that one. In other words, you might have to make a bunch of assumptions till you have reduced your problem to simple harmonic oscillations around a steady state that you have found separately.
