WIAS Preprint No. 2536, (2018)

The point charge oscillator: Qualitative and analytical investigations



Authors

  • Schneider, Klaus R.

2010 Mathematics Subject Classification

  • 34C05 34C15 34C27 78A35

Keywords

  • Global phase portrait, closed orbits, homoclinic orbits, amplitude-period relation, Jacobi elliptic function

DOI

10.20347/WIAS.PREPRINT.2536

Abstract

We determine the global phase portrait of a mathematical model describing the point charge oscillator. It shows that the family of closed orbits describing the point charge oscillations has two envelopes: an equilibrium point and a homoclinic orbit to an equilibrium point at infinity. We derive an expression for the growth rate of the primitive perod Τα of the oscillation with the amplitude α as α tends to infinity. Finally, we determine an exact relation between period and amplitude by means of the Jacobi elliptic function cn.

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