# Is the trajectory of a particle with constant velocity (though its direction can change by collisions) always non-chaotic?

Suppose we have a particle that travels with constant velocity, without heat losses by friction, and no forces acting on it except for occasionally collisions with much bigger wall-like masses than the particle (so that doesn't change the magnitude of its velocity but only its direction). Now, the trajectory of such a particle varies linearly when changing the direction of the velocity by a very small amount. The difference between the two trajectories increases linearly in time.

Is the movement of a particle with a constant velocity (without forces acting on it except at the "moments" it collides with a straight wall, without changing the magnitude of the particle's velocity) always non-chaotic (i.e. varying the initial direction of the particle's velocity gives a linear change in its trajectory) or are there cases that the particle's trajectory is chaotic? For example, if the heavy masses the particle collides with are not straight walls (like a particle moving inside a circular confinement). Or is the trajectory of a single particle (under the aforementioned conditions) always non-chaotic?

## 2 Answers

If by ‘chaotic’ you mean ‘exponentially sensitive to initial conditions’, it’s hard to find a multi-body collisional process that’s not chaotic.

Any process that’s involving i.e. collisions with hard balls or sharp corners will have trajectories where a small displacement $\epsilon$ will make the difference between a hit or a miss. And the number of those, and the trajectory divergence they cause, grows with time.

What you describe is known as a billiard in Nonlinear Dynamics. And it's pretty easy to get chaotic dynamics in billiards:

• if you have dispersing walls (e.g., Sinai's, below, left)
• if you have focusing wall (e.g., Bunimovich's, below, right)

Also geodesic flows (a generalization of the situation you describe) can be chaotic in the presence of negative curvature.

The Scholarpedia entry on billiards is very informative.