On a linearly damped 2 body problem

TL;DR

This paper investigates the dynamics of a linearly damped two-body problem, revealing that particles with non-zero momentum do not reach the center in finite time and all bounded solutions tend to zero asymptotically, with potential special spiraling orbits.

Contribution

It introduces the analysis of damping effects on classical two-body problems, showing non-finite-time approach to the center and asymptotic convergence of solutions.

Findings

Particles with non-zero momentum do not reach the center in finite time.

All bounded solutions tend to zero as time approaches infinity.

Potential existence of special spiraling orbits with exponential convergence.

Abstract

The usual equation for both motions of a single planet around the sun and electrons in the deterministic Rutherford-Bohr atomic model is conservative with a singular potential at the origin. When a dissipation is added, new phenomena appear. It is shown that whenever the momentum is not zero, the moving particle does not reach the center in finite time and its displacement does not blow-up either, even in the classical context where arbitrarily large velocities are allowed. Moreover we prove that all bounded solutions tend to $0$ for $t$ large, and some formal calculations suggest the existence of special orbits with an asymptotically spiraling exponentially fast convergence to the center.