Prograde burn and retrograde burn are quite basic maneuvers in orbital mechanics. But how did the scientists (or researchers) discover that (the mathematical interpretation)? Did they try any practical experiments or was it all a theoretical achievement? How did they discover orbits in the first place?

• Well, to be fair, scientists were aware of the orbits of, say, the moon, and moons of other planets. Observation at least provided support for the fundamental laws of stable orbit. – Carl Witthoft Jan 8 '19 at 15:16