Hidden Symmetries of the Kepler Problem and Jacobi's Theorem
Abstract
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.
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BibTeX
@misc{diaz2026hidden,
title = {{Hidden Symmetries of the Kepler Problem and Jacobi's Theorem}},
author = {Alejandro Moreno Díaz and Juan B. Sancho de Salas},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04791},
eprint = {2611.04791},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04791},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hidden Symmetries of the Kepler Problem and Jacobi's Theorem
AU - Alejandro Moreno Díaz
AU - Juan B. Sancho de Salas
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04791
UR - https://arxiv.org/abs/2611.04791
AB - 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.
ER -