In order to have dynamics, you need to be able to describe rates of change, and to do that, you really need derivatives...and therefore calculus. Calculus was invented by Newton (and Leibniz). Do note that ideas of calculus were already around in Egypt, India and China, but it wasn’t formalised until the 17th century.
(Actually I need to add a very important caveat here. One doesn’t actually need calculus to study dynamical systems. Instead of differential equations, one can use difference equations. I’ve written a short but important post here about them.)
It was at this point that we (and when I say we, I mean Newton) could start writing down differential equations to describe the world. One of the first things that he wanted to understand was gravity. The world already had a set of laws, thanks to Kepler, about planetary motion, and in fact those laws were thanks to the observations of Tycho Brahe - the astronomer with the metal nose (he lost the original in a sword fight). Note importantly that Tycho’s sister, Sophia Brahe also made important contributions to these observations.
The first law: A planet’s orbit around the sun is an ellipse. The sun’s centre is at one focus of the ellipse
This is a pretty nice picture that shows how well the third law holds in the solar system:
So, Brahe had studied the movements of the planets and noted down their positions over time with huge detail. We should also note that Chinese astronomy was way ahead of its time, with Shen Kuo (沈括) having made detailed observations of planetary motion in the 11th century. Kepler had made some empirical observations of these positions and come up with his three laws, but nobody knew why these laws held. So, Newton came along with his new-fangled-fancy-pants calculus, wrote down the differential equation thanks to his law of gravitation and laws of motion, solved them, and proved Kepler's laws to be true (well, at least in the limit that the sun is much much heavier than the planets). You can find a nice review of this here. Quoting from this page:
Kepler's Laws are wonderful as a description of the motions of the planets. However, they provide no explanation of why the planets move in this way. Moreover, Kepler's Third Law only works for planets around the Sun and does not apply to the Moon's orbit around the Earth or the moons of Jupiter. Isaac Newton (1642-1727) provided a more general explanation of the motions of the planets through the development of Newton's Laws of Motion and Newton's Universal Law of Gravitation.
The name of this problem is the so-called two-body problem of celestial mechanics. In simple terms it is the problem of two massive objects (stars, planets, etc.) moving around each other through gravitational interactions. In fact, whenever you have two objects moving about each other, both of them move (the heavier one by less than the lighter one). In this demonstration from here you can explore what happens when you change various parameters in the orbits of two objects around each other. The following is just a picture.
I said that Newton “solved” the two-body problem. What do we mean by solved?
We can write down a set of equations that encodes the dynamics of this system. That is half the battle. They will take the form of differential equations. Once you have the equations, you need to find solutions to the equations. A solution will correspond to a function, or functions of time (the position of the planet and star as a function of time) which satisfy the equation/equations. Generally when we say that someone “solved” a dynamical system, we mean that they found all of the solutions.
How about if you have two planets going around a star? This is the so-called three-body problem. It turns out that this cannot be solved. This means that you cannot write all solutions down in a finite number of ‘normal’ mathematical functions. You can find some very special solutions...but a general solution is not possible...provably! This is an example of a very special solution.