自由点粒子牛顿方程的最大不变性群
The Maximal Invariance Group of Newtons's Equations for a Free Point Particle
摘要：
对自由非相对论性点粒子，牛顿方程的最大不变性群大于Galilei群。它是静态（九参数）Galilei群和包含时间平移、膨胀和单参数时间依赖标度群（称为{ \it展开}）的$SL(2,R)$群的半直积。这个群最早是由Niederer在自由薛定谔方程中发现的。我们还提供了一个路线图，从自由的非相对论性点粒子的流体力学方程的对称性结转。流体力学对称群中迄今未被注意到的$SL(2, R)$部分给出了超新星爆炸数值模拟和惯性约束等离子体中激光诱导内爆实验数值模拟之间观察到的相似性的理论解释。我们也给出了具有这种对称群的点粒子的相互作用的多体系统的例子。
英文摘要：
The maximal invariance group of Newton's equations for a free nonrelativistic point particle is shown to be larger than the Galilei group. It is a semi-direct product of the static (nine-parameter) Galilei group and an $SL(2,R)$ group containing time-translations, dilations and a one-parameter group of time-dependent scalings called {\it expansions}. This group was first discovered by Niederer in the context of the free Schrödinger equation. We also provide a road map from the free nonrelativistic point particle to the equations of fluid mechanics to which the symmetry carries over. The hitherto unnoticed $SL(2, R)$ part of the symmetry group for fluid mechanics gives a theoretical explanation for an observed similarity between numerical simulations of supernova explosions and numerical simulations of experiments involving laser-induced implosions in inertial confinement plasmas. We also give examples of interacting many body systems of point particles which have this symmetry group.
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