Tangent bundle geometry from dynamics: application to the Kepler problem

TL;DR

This paper explores how tangent bundle structures can be derived from dynamical systems on manifolds, specifically applying the framework to the Kepler problem and related harmonic oscillator systems.

Contribution

It introduces a method to identify tangent bundle structures from dynamical vector fields, with a focus on conformal vector fields and the Kepler problem.

Findings

Derived tangent bundle structures for specific dynamical systems

Applied the method to the Kepler problem and harmonic oscillators

Provided a geometric interpretation of Kepler dynamics

Abstract

In this paper we consider a manifold with a dynamical vector field and inquire about the possible tangent bundle structures which would turn the starting vector field into a second order one. The analysis is restricted to manifolds which are diffeomorphic with affine spaces. In particular, we consider the problem in connection with conformal vector fields of second order and apply the procedure to vector fields conformally related with the harmonic oscillator (f-oscillators) . We select one which covers the vector field describing the Kepler problem.