# Revisiting the Kepler problem with linear drag using the blowup method and normal form theory

## TL;DR

This paper analyzes the Kepler problem with linear drag, revealing invariant manifolds and stable sets using blowup and normal form theory, and providing a geometric understanding of the dynamics with dissipation.

## Contribution

It introduces a novel application of blowup and normal form theory to identify invariant manifolds and stable sets in the dissipative Kepler problem, including a zero eccentricity manifold.

## Key findings

- Invariant manifolds are smooth and nonhyperbolic.
- A zero limiting eccentricity manifold is characterized as a center manifold.
- The blowup analysis offers a geometric view of the dissipative dynamics.

## Abstract

In this paper, we revisit the Kepler problem with linear drag. With dissipation, the energy and the angular momentum are both decreasing, but in \cite{margheri2017a} it was shown that the eccentricity vector has a well-defined limit in the case of linear drag. This limiting eccentricity vector defines a conserved quantity, and in the present paper, we prove that the corresponding invariant sets are smooth manifolds. These results rely on normal form theory and a blowup transformation, which reveals that the invariant manifolds are (nonhyperbolic) stable sets of (limiting) periodic orbits. Moreover, we identify a separate invariant manifold which corresponds to a zero limiting eccentricity vector. This manifold is obtained as a generalized center manifold over the zero eigenspace of a zero-Hopf point. Finally, we present a detailed blowup analysis, which provides a geometric picture of the dynamics. %We will use this to shed light on a degenerate case of the limiting eccentricity vector. %The aforementioned invariant manifolds only make up a subset hereof. We believe that our approach and results will have general interest in problems with blowup dynamics.