The point charge oscillator: Qualitative and analytical investigations
- Schneider, Klaus R.
2010 Mathematics Subject Classification
- 34C05 34C15 34C27 78A35
- Global phase portrait, closed orbits, homoclinic orbits, amplitude-period relation, Jacobi elliptic function
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.