classical mechanics - Motion of a bouncing sphere with a spring attached inside

Imagine a sphere with inside a spring attached (between opposite sides). You let it fall from a certain height, after which it bounces from a flat surface. The sphere is rigid.

Will the following motion be chaotic or can we put a nice function upon its subsequent motion? For simplicity, we’ll let friction out of the game.

Answer

I assume that the spring’s mass is distributed homogeneously across the spring.

In this case, you have to use a partial differential equation (PDE) to perfectly describe the spring, whose solution would for example be a function which describes how each infinitesimal segment of the spring is shifted from its resting position (when no external forces and gravity are applied). Under benign boundary conditions the solution to these PDE would be longitudinal waves propagating on the spring.

However, the boundary conditions of the PDE for this problem are described by the sphere’s motion but also feed back into the sphere’s motion: The spring perceives a “kick” (starting a new wave) whenever the sphere bounces, and the sphere’s motion is affected by waves reaching the end of the spring. Thus I strongly doubt that there is a nice analytic solution to this problem – except perhaps in some benign cases, such as when the spring’s natural frequency matches the frequency of the bouncing.

As for chaoticity, you have infinitely many degrees of freedom (the position and momentum of the sphere as well as that of every point in the spring). You also have at least one clear non-linear component introduced by the sphere’s bouncing off the surface (you can have more if the spring is non-linear or due to the PDE’s boundary conditions). So, all the necessary ingredients for chaos are present.

I thus would expect chaos for at least some parameter choices. While I cannot prove this system without actually simulating it, I can appeal to intuition: If the parameters of the system are such, that the reflection of a spring wave at the spring’s end can be seen as a singular event, it can make a considerable difference whether this reflection happens before or after the sphere bouncing. Here a tiny difference in the state can eventually considerably affect the dynamics and thus you have the butterfly effect. Moreover the system bears a certain resemblance to the kicked oscillator, which exhibits chaos and is arguably less complicated.