论文标题
可集成玻尔兹曼系统的Poncleet属性和准周期性
Poncelet property and quasi-periodicity of the integrable Boltzmann system
论文作者
论文摘要
我们研究平面中粒子的运动，受到有吸引力的中央力，并在壁的一侧具有反方定律，其弹性反射。该模型是Boltzmann考虑的一类系统的特殊情况，该系统最近由Gallavotti和Jauslin表明，该系统承认了能量外的第二个运动积分。通过记录粒子撞击墙壁时的后续位置和动量，我们获得了三维离散时间动力系统。我们表明该系统具有poncleet属性：如果对于积分的给定通用值，一个轨道是周期性的，那么这些值的所有轨道都是周期性的并且具有相同的时期。之所以与poncleet定理：运动积分的通用水平集是椭圆曲线相同，庞加莱映射是两个带有固定点的两个相关的组成，因此是固定元素的翻译。我们结果的另一个结果是证明了Gallavotti和Jauslin在可集成的玻尔兹曼系统的准周期性上的猜想，这表明KAM扰动理论对具有弱离心力的Boltzmann系统的适用性。
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, 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.
