Show using (6.201) that \(d \mathbf{A} / d t=0\). Draw an elliptical orbit in the \(x-y\) plane and show on it the Laplace-Runge-Lenz vector \(\mathbf{A}\). The existence of this conserved vector quantity is the reason why one can smoothly deform ellipses into a circle without changing the energy of the system. Mathematically, this additional hidden symmetry implies that the Newtonian problem is equivalent to a free particle on a three-dimensional sphere embedded in an abstract four-dimensional world. It is believed that this is a mathematical accident; no physical significance of this fourth dimension has yet been identified...

Data from eq. 6.201

Avx (rxv) - k. r