Dynamical quasi-symmetry of the classical Kepler-Coulomb problem and the conservation of the Laplace-Runge-Lenz vector

Abstract

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.