To get really picky, you only need 2 objects, not three. Having a planet going around a star makes things more “interesting” but you can still achieve the same affect with just a planet and a space probe.
Note that it is a refractive collision, rather than a standard reflective one that we are familiar with.
Image a fast moving ball hitting a stationary block of wood that is tilted at an angle. The block moves away and the ball bounces off in another direction. Since the block is now moving, the ball has lost energy and is thus moving slower. If you take a film of the collision and run it backwards, you see the slowly moving block and the probe colliding, the ball picks up speed.
That’s a reflection, gravity wells are refractors. But you are still essentially “hitting” something. Depending on the relative directions and such you can either speed up or slow down the probe.
So in effect, there is no difference in looking at gravity assist problems from a Newtonian or GR point of view, aside from a few minor calculations in precision.
They both essentially explain gravity assist scenarios from equivalent perspectives.
Well, as long as you’re not dealing with black holes, or relativistic jets, or nanosecond precision, then GR is always essentially the same as Newtonian mechanics. One aspect of celestial dynamics in GR is that in weak gravity fields and low speeds, it reduces to classical dynamics.
Note that differences between Newtonian physics and GR have been measured in the orbit of Mercury. Close to the Sun and elliptical orbit. People spend a lot of effort trying to reconcile its orbit with GR, trying to squeeze out the last bit of error in measurements vs. formulas.
For the rest of the solar system, GR differences are quite low order effects.