The classical dynamic symmetry for the \(\mathrm{U}(1)\)-Kepler problems (Q1742261)

It has been known for more than 40 years that the Lie algebra \(\mathfrak{so}(2, 4)\) has a Poisson realization on \(M=T^*\mathbb{R}^3_*\) from which we can reproduce the Kepler problem, i.e., the mathematical model of the simplest solar system. Here \(\mathbb{R}^3_\prime=\mathbb{R}^3\setminus \{0\}\) is the configuration space for the Kepler problem. In Section 4 of the present paper, a family of Poisson realizations for \(\mathfrak{su}(n, n)\) is presented in Theorem 4.2.1. As a consequence of this result a Laplace-Runge-Lenz (LRL) vector for any \(\mathrm{U}(1)\)-Kepler problem is obtained. As corollary, a formula connecting its Hamiltonian to its angular momentum and its RLR vector is derived.