The classical dynamic symmetry for the $\mathrm{Sp}(1)$-Kepler problems

TL;DR

This paper constructs a Poisson realization of the Lie algebra a9*(4n) on the phase space of a9(1)-Kepler problems, revealing the classical Laplace-Runge-Lenz vector and simplifying the verification process using Weinstein's idea.

Contribution

It introduces a new Poisson realization framework for a9*(4n) in a9(1)-Kepler problems, linking it to the canonical realization on a specific Poisson manifold.

Findings

Poisson realization of a9*(4n) established for a9(1)-Kepler problems

Laplace-Runge-Lenz vector derived for these problems

Simplified verification via Weinstein's idea

Abstract

A Poisson realization of the simple real Lie algebra $\mathfrak {so}^*(4n)$ on the phase space of each $\mathrm {Sp}(1)$-Kepler problem is exhibited. As a consequence one obtains the Laplace-Runge-Lenz vector for each classical $\mathrm{Sp}(1)$-Kepler problem. The verification of these Poisson realizations is greatly simplified via an idea due to A. Weinstein. The totality of these Poisson realizations is shown to be equivalent to the canonical Poisson realization of $\mathfrak {so}^*(4n)$ on the Poisson manifold $T^*\mathbb H_*^n/\mathrm{Sp}(1)$. (Here $\mathbb H_*^n:=\mathbb H^n\backslash \{0\}$ and the Hamiltonian action of $\mathrm{Sp}(1)$ on $T^*\mathbb H_*^n$ is induced from the natural right action of $\mathrm{Sp}(1)$ on $\mathbb H_*^n$. )