Abstract :
We consider the class of dynamical systems $\ddot{\boldsymbol{x}}=r^{-3}I_1 \boldsymbol{x}+r^{-2}I_2 \dot{\boldsymbol{x}}$, $\boldsymbol{x} \in R^{3}$, $r=\left| \boldsymbol{x} \right|$, where $I_1, I_2$ are each functions of $\boldsymbol{x}$ and $\dot{\boldsymbol{x}}$ and invariants of the nonlocal symmetry transformation $\bar{\boldsymbol{x}}=h^{-1}\boldsymbol{x}$, ${d\bar{t} \mathord{\left/{\vphantom{d\bar{t}dt}}\right.\kern-\nulldelimiterspace} dt} = h^{-2}$, $h=1+\boldsymbol{q.x}$ and $\boldsymbol{q}$ is any constant vector. It is shown in a recent paper by the author, that this equation is the most general form of the (second-order) dynamical system which admits the above transformation as a symmetry transformation. We show that for $I_2 \ne 0$ such systems possess (a) a constant of motion $w(\alpha,L)$ which satisfies the equation $\partial_{\alpha}w+I_2 \partial_{L}w=0$, where $(r,\alpha)$ are polar coordinates in the plane of motion, $L=\left|\boldsymbol{L} \right|$, $\boldsymbol{L} = \boldsymbol{x} \wedge \dot{\boldsymbol{x}}$ and any other such function is some function of $w$ (b) a Laplace-Runge-Lenz vector constant of motion $\boldsymbol{J}$ given by $\boldsymbol{J}=\hat{\boldsymbol{L}} \wedge \boldsymbol{K}$ where $\boldsymbol{K}=L^{-1}\dot{\boldsymbol{x}}+\boldsymbol{W}$, $\hat{\boldsymbol{L}}=L^{-1} \boldsymbol{L}$ and $\boldsymbol{W}$ is a function of $w$ and $\alpha$ with $\boldsymbol{L.W}=0$. We compute the Ermanno-Bernoulli constants and use them to obtain nonlocal symmetry transformation of such systems. We then use the analysis of K. Andriopoulos and P.G.L. Leach (2002) on minimal generating sets of symmetry vectors which specify the harmonic oscillator to obtain complete symmetry groups for these systems.
