You see different types of funnels.
The profile of the first one resembles a hyperbolic
curve, the second is a half-sphere,
the third resemble a parabolic curve.
Try to launch a ball inside these funnels (click on the photo). As you see, the ball performs
quite a complex trajectory, swinging in different directions.
Only for the first (hyperbolic shape) funnel
the trajectory is closed.
This funnel (to a rough approximation) models the gravitational field, changing with distant r
like 1/r -
the one that makes Earth orbiting around Sun.
This is an important observation: only a 1/r field
(and also r2, like in a harmonic oscillator -
a spring) assures closed orbits.
Imagine
what would happen with seasons on Earth if the gravitational field would be
different:
summer coming every year in different month!
It was known from mid of
XIX century, that Mercury planet performs an
open orbit.
Einstein in 1916 explained why: the field is slighly different from 1/r,
due to a huge mass of Sun (which changes the geometry of Space).
This was one of the proofs for General Theory of Relativity. We
can simmulate different fields by this programme
(courtesy of dott. Mauro Brunelli).
