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arXiv · 2303.00283

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

Kristian Uldall Kristiansen
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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.

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Kristian Uldall Kristiansen. 2023-03-01. Revisiting the Kepler problem with linear drag using the blowup method and normal form theory. https://arxiv.org/abs/2303.00283

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