经典开普勒-库仑问题的动力学拟对称性与拉普拉斯-龙格-楞次矢量的守恒
Dynamical quasi-symmetry of the classical Kepler-Coulomb problem and the conservation of the Laplace-Runge-Lenz vector
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中文总结 AI 辅助
本文证明经典开普勒-库仑问题中存在一种射影变换与时间标度构成的动力学拟对称性，并由此推导出拉普拉斯-龙格-楞次矢量的守恒。
中文摘要 AI 辅助
本文研究在 $\mathbb{R}^{N}$（$N\geqslant2$）中，受开普勒-库仑力 $\boldsymbol{F}(\boldsymbol{r})=\alpha\boldsymbol{r}/r^{3}$（$\alpha\neq0$）作用的经典非相对论性粒子。证明了扩展构型空间变换——包含径向矢量 $\boldsymbol{r}\to\boldsymbol{R}=\boldsymbol{r}/(1+\boldsymbol{c}\cdot\boldsymbol{r})$ 的射影变换和微分时间标度 $\mathrm{d}t\to\mathrm{d}T=\mathrm{d}t/(1+\boldsymbol{c}\cdot\boldsymbol{r})^{2}$，其中 $\boldsymbol{c}$ 为具有逆长度量纲的实矢量参数——构成一种动力学拟对称性，其意义在于将牛顿运动方程 $m(\mathrm{d}^{2}\boldsymbol{r}/\mathrm{d}t^{2})=\alpha\boldsymbol{r}/r^{3}$ 映射为方程 $m(\mathrm{d}^{2}\boldsymbol{R}/\mathrm{d}T^{2})=(\alpha\boldsymbol{R}/R^{3}) \operatorname{sgn}(1-\boldsymbol{c}\cdot\boldsymbol{R})$。该拟对称性独立于且互补于众所周知的、对负能量为 SO($N+1$) 和对正能量为 SO($N,1$) 的开普勒-库仑动力学对称性。文中对原始系统和变换系统相关联的非退化圆锥轨迹之间诱导映射进行了详细的解析和图形分析，并简要讨论了速度星形图之间的映射。随后证明了变换运动中能量的守恒蕴含原始动力学中拉普拉斯-龙格-楞次矢量的守恒。
英文摘要
The paper studies a classical non-relativistic particle moving in $\mathbb{R}^{N}$, $(N\geqslant2)$, in the presence of the Kepler-Coulomb force $\boldsymbol{F}(\boldsymbol{r})=α\boldsymbol{r}/r^{3}$, $(α\neq0)$. It is shown that the extended-configuration-space transformation involving the projective transformation of the radius vector $\boldsymbol{r}\to\boldsymbol{R}=\boldsymbol{r}/(1+\boldsymbol{c}\cdot\boldsymbol{r})$ and the differential time scaling $\mathrm{d}t\to\mathrm{d}T=\mathrm{d}t/(1+\boldsymbol{c}\cdot\boldsymbol{r})^{2}$, where $\boldsymbol{c}$ is a real vector-valued parameter with dimensions of inverse length, constitutes a dynamical quasi-symmetry in the sense that it maps Newton's equation of motion $m(\mathrm{d}^{2}\boldsymbol{r}/\mathrm{d}t^{2})=α\boldsymbol{r}/r^{3}$ into the equation $m(\mathrm{d}^{2}\boldsymbol{R}/\mathrm{d}T^{2})=(α\boldsymbol{R}/R^{3}) \operatorname{sgn}(1-\boldsymbol{c}\cdot\boldsymbol{R})$. This quasi-symmetry is independent of, yet complementary to, the well-known Kepler-Coulomb dynamical symmetry, which is SO($N+1$) for negative energies and SO($N,1$) for positive energies. A detailed analytical and graphical analysis of induced mappings between non-degenerate conic trajectories associated with the original and transformed systems is presented. Mappings between velocity hodographs are also briefly discussed. It is then proved that the conservation of energy in the transformed motion implies the conservation of the Laplace-Runge-Lenz vector in the original dynamics.
发表机构
- Gdańsk University of Technology（格但斯克理工大学）
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