Perturbed $(2n-1)$-Dimensional Kepler Problem and the Nilpotent Adjoint Orbits of $U(n, n)$

TL;DR

This paper explores the regularization of the high-dimensional Kepler problem and its relation to nilpotent coadjoint orbits of a unitary group, proposing integrable perturbations and analyzing regularization methods.

Contribution

It introduces a unified view of regularization procedures and proposes new integrable perturbations of the high-dimensional Kepler problem.

Findings

Equivalence of Kustaanheimo-Stiefel and Cayley regularizations.

Proposal of integrable generalizations of the $(2n-1)$-Kepler problem.

Analysis of the relation between Hamiltonian systems and nilpotent coadjoint orbits.

Abstract

We study the regularized $(2n-1)$-Kepler problem and other Hamiltonian systems which are related to the nilpotent coadjoint orbits of $U(n,n)$. The Kustaanheimo-Stiefel and Cayley regularization procedures are discussed and their equivalence is shown. Some integrable generalization (perturbation) of $(2n-1)$-Kepler problem is proposed.