# On a paradox in the impact dynamics of smooth rigid bodies

## TL;DR

This paper reveals a paradox where, even without friction, two smooth convex rigid bodies can interpenetrate after collision, challenging the compatibility of rigid body dynamics with classical conservation laws.

## Contribution

It demonstrates that rigid body impact dynamics can lead to interpenetration without friction, highlighting fundamental inconsistencies in classical mechanics assumptions.

## Key findings

- Interpenetration occurs in both 2D and 3D collisions.
- Penetration distance scales as O(t^3).
- Rigid body dynamics conflicts with conservation laws.

## Abstract

Paradoxes in the impact dynamics of rigid bodies are known to arise in the presence of friction. We show here that, on specificc occasions, in the absence of friction, the conservation laws of classical mechanics are also incompatible with the collisions of smooth, strictly convex rigid bodies. Under the assumption that the impact impulse is along the normal direction to the surface at the contact point, two convex rigid bodies which are well separated can come into contact, and then interpenetrate each other. This paradox can be constructed in both 2D and 3D when the collisions are tangential, in which case no momentum or energy transfer between the two bodies is possible. The postcollisional interpenetration can be realized through the contact points or through neighboring points only. The penetration distance is shown to be O(t^3). The conclusion is that rigid body dynamics is not compatible with the conservation laws of classical mechanics.