Particle Dynamics in Time-Dependent Coulomb Field

J. Cronvall, T.S. Turova

2021, v.27, Issue 2, 197-227

ABSTRACT

We study a non-autonomous differential equation describing the one-dimensional motion of a particle interacting with a charged source at the origin. The charge of the source is piece-wise constant; it alternates between two different values, one being positive and the other being negative. We show that the alternation of the charge enriches the behaviour of the system, exhibiting qualitatively new features.
In particular, we find necessary and sufficient conditions for the parameters of the model which yield periodic bounded solutions, and we describe some necessary conditions as well.
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The states of the particle at the moments of switches define a natural embedded discrete dynamical system.
We find the fixed points of this dynamical system and describe conditions for their stability.
The 2-periodic points are described in detail.

Keywords: Coulomb mechanics, time-dependent force, stable cycles, point particles

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