Parametric Resonance of a Charged Pendulum with a Suspension Point Oscillating Between Two Vertical Charged Lines

Abstract

In this study, we analyze a planar mathematical pendulum with a suspension point that oscillates harmonically in the vertical direction. The bob of the pendulum is electrically charged and is located between two wires with a uniform distribution of electric charges, both equidistant from the suspension point. The dynamics of this phenomenon is investigated. The system has three parameters, and we analyze the parametric stability of the equilibrium points, determining surfaces that separate the regions of stability and instability in the parameter space. In the case where the parameter associated with the charges is equal to zero, we obtain boundary curves that separate the regions of stability and instability for the Mathieu equation.

APPENDIX A. BOUNDARY SURFACES FOR THE EQUILIBRIUM $$P_{1}$$ WHEN $$N=4,5,6$$

Here, we present the boundary surface parameterizations for the $P_{1}$ equilibrium associated with the resonances $2\omega=N$, where $N=4,5,6$. To do so, we use the Hamiltonian (3.7) with $\omega_{0}^{2}=\frac{\mu}{4}+\alpha_{0}$.

• For $N=4$

  $$\displaystyle\alpha=\frac{16-\mu}{4}+\frac{1}{30}\varepsilon^{2}+\frac{433}{216000}\varepsilon^{4}-\frac{5701}{170100000}\varepsilon^{6}+\mathcal{O}(\varepsilon^{7}),$$

  (A.1)
  $$\displaystyle\alpha=\frac{16-\mu}{4}+\frac{1}{30}\varepsilon^{2}-\frac{317}{216000}\varepsilon^{4}+\frac{4799}{170100000}\varepsilon^{6}+\mathcal{O}(\varepsilon^{7}).$$

  (A.2)

• For $N=5$

  $$\alpha=\frac{25-\mu}{4}+\frac{1}{48}\varepsilon^{2}+\frac{11}{193536}\varepsilon^{4}\mp\frac{1}{18432}\varepsilon^{5}+\frac{37}{55738368}\varepsilon^{6}+\mathcal{O}(\varepsilon^{7}).$$

  (A.3)

• For $N=6$

  $$\displaystyle\alpha=\frac{36-\mu}{4}+\frac{1}{70}\varepsilon^{2}+\frac{187}{10976000}\varepsilon^{4}+\frac{6743617}{5808499200000}\varepsilon^{6}+\mathcal{O}(\varepsilon^{7}),$$

  (A.4)
  $$\displaystyle\alpha=\frac{36-\mu}{4}+\frac{1}{70}\varepsilon^{2}+\frac{187}{10976000}\varepsilon^{4}-\frac{5861633}{5808499200000}\varepsilon^{6}+\mathcal{O}(\varepsilon^{7}).$$

  (A.5)

APPENDIX B. BOUNDARY SURFACES FOR THE EQUILIBRIUM $$P_{2}$$ WHEN $$N=4,5,6$$

Similarly to what was done for the equilibrium $P_{1}$, we present the boundary surface parameterizations for the equilibrium $P_{2}$ associated with the resonances $2\omega=N$, where $N=4,5,6$. In this case, we use the Hamiltonian (3.7) with $\omega_{0}^{2}=\frac{\mu}{4}-\alpha_{0}$.

• $N=4$

  $$\displaystyle\alpha=\frac{\mu-16}{4}-\frac{1}{30}\varepsilon^{2}-\frac{433}{216000}\varepsilon^{4}+\frac{5701}{170100000}\varepsilon^{6}+\mathcal{O}(\varepsilon^{7}),$$

  (B.1)
  $$\displaystyle\alpha=\frac{\mu-16}{4}-\frac{1}{30}\varepsilon^{2}+\frac{317}{216000}\varepsilon^{4}-\frac{4799}{170100000}\varepsilon^{6}+\mathcal{O}(\varepsilon^{7}).$$

  (B.2)

• $N=5$

  $$\alpha=\frac{\mu-25}{4}-\frac{1}{48}\varepsilon^{2}-\frac{11}{193536}\varepsilon^{4}\mp\frac{1}{18432}\varepsilon^{5}-\frac{37}{55738368}\varepsilon^{6}+\mathcal{O}(\varepsilon^{7}).$$

  (B.3)

• $N=6$

  $$\displaystyle\alpha=\frac{\mu-36}{4}-\frac{1}{70}\varepsilon^{2}-\frac{187}{10976000}\varepsilon^{4}-\frac{6743617}{5808499200000}\varepsilon^{6}+\mathcal{O}(\varepsilon^{7}),$$

  (B.4)
  $$\displaystyle\alpha=\frac{\mu-36}{4}-\frac{\varepsilon^{2}}{70}-\frac{187\varepsilon^{4}}{10976000}+\frac{5861633\varepsilon^{6}}{5808499200000}+\mathcal{O}(\varepsilon^{7}).$$

  (B.5)