Runge-Lenz Vector as a 3d Projection of SO(4) Moment Map in ${\mathbb{R}}^4\times {\mathbb{R}}^4$ Phase Space

We show, using the methods of geometric algebra, that Runge-Lenz vector in the Kepler problem is a 3-dimensional projection of SO(4) moment map that acts on the phase space of 4-dimensional particle motion. Thus, RL vector is a consequence of geometric symmetry of ${\mathbb{R}}^4\times {\mathbb{R}}^4$ phase space.