经典力学中的动力学对称性--Laplace-Runge-Lenz矢量再探
Illustrating Dynamical Symmetries in Classical Mechanics: The Laplace-Runge-Lenz Vector Revisited
摘要：
平方反比力定律允许有一个位于运动平面内的守恒矢量.这个矢量与拉普拉斯、龙格和伦茨等名字有关。许多工作者已经探索了与这个矢量和类似的动力学矢量相关的对称性和简并性。我们定义了一个保守的动力学变量$α$，它表征了开普勒问题的二维位形空间中的轨道方向，并为各向同性谐振子定义了一个类似的变量$β$。这个轨道方向变量与垂直于运动平面的角动量分量正则共轭。我们探讨了由$α(β).$产生的正则单参数变换群。由于我们有一对明显的守恒正则共轭变量，所以我们希望将它们作为坐标-动量对来使用。根据这些相空间坐标，哈密顿量的形式几乎是平凡的，因为这对中的任何一个成员都不能显式地出现在哈密顿量中。从这些考虑，我们得到一个简单的图片相空间中的动态。我们使用的程序是在汉密尔顿-雅可比方法的精神。
英文摘要：
The inverse square force law admits a conserved vector that lies in the plane of motion. This vector has been associated with the names of Laplace, Runge, and Lenz, among others. Many workers have explored aspects of the symmetry and degeneracy associated with this vector and with analogous dynamical symmetris. We define a conserved dynamical variable $α$ that characterizes the orientation of the orbit in two-dimensional configuration space for the Kepler problem and an analogous variable $β$ for the isotropic harmonics oscillator. This orbit orientation variable is canonically conjugate to the angular momentum component normal to the plane of motion. We explore the canoncial one-parameter group of transformations generated by $α(β).$ Because we have an obvious pair of conserved canonically conjugate variables, it is desirable to us them as a coordinate-momentum pair. In terms of these phase space coordinates, the form of the Hamiltonian is nearly trivial because neither member of the pair can occur explicitly in the Hamiltonian. From these considerations we gain a simple picture of the dynamics in phase space. The procedure we use is in the spirit of the Hamilton-Jacobi method.
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