Title: Weyl Group Representation of Billiard Trajectories for One-dimensional Hard Sphere Dynamics Show Chinese title

Title: 一维硬球动力学中弹道轨迹的Weyl群表示

Abstract: 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. Show Chinese abstract

Abstract: 我们给出了一个精确公式，用于描述半径为$r>0$的$N$个硬球在无限长直线上动力学行为的演化过程，假设系统的总动量和动能在所有时刻都守恒。 这个模型通常被称为一维Tonks气体或硬杆气体模型。 我们的精确公式表示为与根系$A_{N-1}$对应的Weyl群的求和，并且对于硬球桌切丛的一个满测度子集中的所有初始数据均有效。 作为我们显式公式的应用，我们给出了一种简单的证明：即硬球桌切丛上的关联散射流具有Liouville测度作为不变测度。