in simplification, you could derive that curvature of the orbit is basically the force between objects, with radius, from the centre of mass. Once you know curvature at a point, you ca define a series of curves fitting, and substitute with other postions to find all constants. And what is curvature? you can find online, that curvature is just a function of second order derivative, and you just have to solve a second degree differential equation.

Orbit is esspetially free falling past the side of the planet, fast enough that the force of gravity doesn't completely pull you in. So you kind of fall past the planet, but get curved towards it enough you don't go flying out away either. The minimum velocity to orbit around a planet with radius r and gravity g is Vorbit = √(rg) I believe

Orbits, it turns out, are mostly just rates of change.

So, the heart of the issue is that each object's path changes continuously, and the forces involved change in kind. Even worse, the objects interact with each other, again continuously - it's not one-sided.

If you imagine trying to do it pre-Calculus, some kind of "just map it all out into a grid, etc.", you can see the problems this continuous change imposes (exercise left for the reader).

By involving the Stravinsky Interpretation, it quickly becomes clear that the dimorphic superposition destabilizes. The clever reader might object "but what if you fold in all the noodly surfaces to recohere the manifold?"

And that clever reader would be right! But we didn't know that until old Dr. Isaac "Zeke" Newton came along and made it that way.

Some say the devil himself taught him how it's done, because no one else can read his notes! So keep your eye on old Zeke when you run into him.

An orbit is analogous to position. Trying to predict the position of planets proved difficult to do accurately. People had figured out velocity a long time ago, but no one had ever explored the concept of describing an orbit just from the accelerations of the planets. It turns out acceleration has a fairly simple equation, and in order to get back to the observable thing of orbits, you need to integrate it twice. Newton was the first person to describe acceleration and to use it to predict the motion of planets. Especially for predicting Mars, it was essential because Jupiter is massive enough and close enough to alter its orbit, so simple Kepler's Laws aren't sufficient to describe the orbit.