The symmetry regularization of 1D generalized Kepler problems

Abstract

The 1D generalized Kepler problem with magnetic charge $\mu \ge 0$ is the Hamiltonian system for which the phase space is the total cotangent space of the half line $R_{+}:=(0,\infty )$ and the Hamiltonian is $H=\frac{1}{2}{p}^2+\frac{{\mu }^2}{2q^2}-\frac{1}{q}$ where $q\in R_{+}$ is the position and $p\in R$ is the momentum. Let ${\Sigma }_{+}$ be the positive portion of the phase space (i.e., $\{H>0\}$) and ${\Sigma }_{-}$ be the negative portion of the phase space. Denote by $O_a$ the co-adjoint orbit of ${\mathrm{sl}}_2\left(R\right)\equiv R^{1,2}$ defined by conditions $x_0^2-x_1^2-x_2^2=a^2$ and $x_0>0$. It is demonstrated that there is a symplectic embedding ${\iota }_{-}$: ${\Sigma }_{-}\to O_{\mu }$ such that the image of ${\iota }_{-}$ is dense and open, and any Kepler motion (i.e. Hamiltonian motion under H) inside ${\Sigma }_{-}$, after being embedded via ${\iota }_{-}$, extends to a complete Hamiltonian motion inside $O_{\mu }$. The positive case is similar, but with $O_{\mu }$ replaced by $O_{\mu /2}$, i.e., there is a symplectic embedding ${\iota }_{+}$: ${\Sigma }_{+}\to O_{\mu /2}$ that satisfies similar conditions. A revelation from the work here is that the term “regularization map” is a misnomer, and the correct name is S-duality, a term that appears in string theory/gauge theory and resembles the Fourier transform.