Weyl Group Representation of Billiard Trajectories for One-dimensional Hard Sphere Dynamics

We present an exact formula for the dynamics of $N$ hard spheres of radius $r>0$ on an infinite line which evolve under the assumption that total linear momentum and kinetic energy of the system is conserved for all times. This model is commonly known as the one-dimensional Tonks gas or the hard rod gas model. Our exact formula is expressed as a sum over the Weyl group associated to the root system $A_{N-1}$ and is valid for all initial data in a full-measure subset of the tangent bundle of the hard sphere table. As an application of our explicit formula, we produce a simple proof that the associated billiard flow admits the Liouville measure on the tangent bundle of the hard sphere table as an invariant measure.