We study the motion of a particle in a plane subject to an attractive central force with inverse-square law on one side of a wall at which it is reflected elastically. This model is a special case of a class of systems considered by Boltzmann which was recently shown by Gallavotti and Jauslin to admit a second integral of motion additionally to the energy. By recording the subsequent positions and momenta of the particle as it hits the wall, we obtain a three-dimensional discrete-time dynamical system. We show that this system has the Poncelet property: If for given generic values of the integrals one orbit is periodic, then all orbits for these values are periodic and have the same period. The reason for this is the same as in the case of the Poncelet theorem: The generic level set of the integrals of motion is an elliptic curve, and the Poincaré map is the composition of two involutions with fixed points and is thus the translation by a fixed element. Another consequence of our result is the proof of a conjecture of Gallavotti and Jauslin on the quasi-periodicity of the integrable Boltzmann system, implying the applicability of KAM perturbation theory to the Boltzmann system with weak centrifugal force.

我们研究了粒子在平面上的运动，该平面在壁的一侧受到中心吸引力的吸引，并具有平方反比定律，在该处弹性反射。该模型是玻尔兹曼所考虑的一类系统的特例，最近由加拉沃蒂（Gallavotti）和贾斯林（Jauslin）证明，该系统除了能量以外还接受第二运动积分。通过记录粒子撞击壁的后续位置和动量，我们获得了三维离散时间动力系统。我们证明该系统具有Poncelet属性：如果对于给定的积分一般值，一个轨道是周期性的，则这些值的所有轨道都是周期性的，并且周期相同。其原因与Poncelet定理相同：运动积分的一般水平集为椭圆曲线，庞加莱地图是由两个具有固定点的对合组成，因此由固定元素进行平移。我们结果的另一个结果是证明了Gallavotti和Jauslin关于可积Boltzmann系统的准周期的猜想，这表明KAM扰动理论对弱离心力的Boltzmann系统的适用性。