Hidden Symmetries of the Kepler Problem and Jacobi's Theorem
We present a geometric derivation of the hidden symmetries of the Kepler problem, combining a duality for central-force systems, obtained by interchanging positions and velocities, with a generalized form of Jacobi's theorem. Applied to the dual Kepler problem, this theorem identifies Kepler hodographs of prescribed energy with geodesics of a metric of constant curvature. After adjoining one point at infinity, the resulting geometry is complete. Its Killing vector fields induce a finite-dimensional Lie algebra of infinitesimal symmetries on the corresponding energy hypersurface. The distinguished point at infinity determines a canonical decomposition of the Killing algebra into two vector subspaces, whose associated Noether invariants recover angular momentum and the Runge-Lenz vector.