On the trajectories of \(\mathrm{U}(1)\)-Kepler problems (Q2825602)

The setting of this paper is provided by the space \(C_1\) of rank-one semi-positive elements in the Euclidean Jordan algebra \(H_n(\mathbb{C})\) of complex Hermitian matrices of order \(n\). There are two canonical structures on \(C_1\): the Kepler metric and the Kepler \(2\)-form \(\omega _K\). For a real number \(\mu \) the author introduces on \(T^{\ast}C_1\) the symplectic form \(\omega _{\mu }:= \omega _{C_1}+2\mu \pi ^{\ast }\omega _K\) where \(\omega _{C_1}\) is the canonical symplectic form of \(T^{\ast}C_1\) and \(\pi \) is the cotangent bundle projection. The symplectic manifold \(M^{\mu }:=(T^{\ast}C_1, \omega _{\mu })\) will serve as the phase space of the \(\mathrm U(1)\)-Kepler problem with magnetic charge \(\mu \). A main result is that a trajectory of this problem is always the intersection of \(C_1\) with a real plane inside \(H_n(\mathbb{C})\) and hence, is a quadratic curve.NEWLINENEWLINEFor the entire collection see [Zbl 1317.00020].