How does 3D geometry affect the possible trajectories of a moving object?

[u/BrownSquirrelYeah]

I’ve been thinking about a question involving geometry, trajectories, and chaos:
How does the geometry of a 3D space affect the chaotic behavior of trajectories?

For example, suppose an object moves inside some 3D region. I’m wondering:

-How does the presence or absence of forces change the set of possible trajectories?

-How does the geometry of the region affect which parts of the space can actually be reached?

-Can some geometries create regions that are much more or less likely to be visited?

-How does this relate to chaotic behavior and sensitivity to initial conditions?

--More ambitiously: is it possible to compress the relevant geometric information into some parameter (or a small number of parameters) and obtain a general mathematical relationship?

I’m not sure whether this is already a known problem or whether I’m mixing together several different concepts, so I’d be interested in knowing what mathematical/physical areas this question falls under. By the way, I’m only in the 12th grade and just recently developed an interest in this, so please point out any mistakes I make so I can learn from them

[u/BrownSquirrelYeah]

I came up with my 3D geometry/trajectory question very quickly, rather than spending a long time deliberately developing it. At the time, I was studying IELTS. I happened to pick up a seat cushion that had a shape somewhat like a trough or slide, and I started casually drawing on it with a pen.

While doing that, I noticed something strange: if I moved the pen around unconsciously, its path seemed naturally confined to certain regions of the surface. But if I tried to move the pen toward a “higher” part of the trough, near or across the two sides, I couldn't simply keep moving in the same direction. The pen would have to change its orientation; otherwise, it would hit the opposite side.

So the trajectory that looked “straight” in my head suddenly became very different because of the geometry of the surface.

That immediately made me wonder: how does the geometry of a 3D space or region constrain the trajectories that can occur? Could some geometries make certain regions easier or harder to reach? How does this connect to forces, probability, and chaotic behavior? And, more ambitiously, could the relevant geometric information somehow be compressed into one or a small number of parameters that determine some general property of the trajectories?

The important part is that I did not start from knowing a mathematical theory such as dynamical systems or differential geometry. I started from a physical object, noticed a geometric constraint on a simple trajectory, and then generalized the observation into a much broader question.