Regularisation by Hamiltonian extension

TL;DR

This paper introduces a unique, real-analytic symplectic extension of phase space for Kepler-like potentials, ensuring complete and real-analytic Hamiltonian flows across arbitrary dimensions.

Contribution

It constructs a novel symplectic extension for Kepler potentials that guarantees flow completeness and real-analyticity in any dimension.

Findings

Established a unique symplectic extension for Kepler potentials

Ensured Hamiltonian flow completeness in the extended phase space

Maintained real-analytic properties of the flow

Abstract

We consider the Kepler potential and its relatives $q\mapsto -\|q\|^{-2(1-1/n)}$, $n\in\mathbb{N}$ in arbitrary dimension $d$. We derive a unique real-analytic symplectic extension of phase space on which the Hamiltonian flow is complete and still real-analytic.