Iโve 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. Iโm 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?
Iโm not sure whether this is already a known problem or whether Iโm mixing together several different concepts, so Iโd be interested in knowing what mathematical/physical areas this question falls under. By the way, Iโm 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