Consider the oscillating Paul trap potential [U(z, ho)=frac{U_{0}+U_{1} cos Omega t}{ho_{0}^{2}+2 z_{0}^{2}}left(2 z^{2}+left(ho_{0}^{2}-ho^{2} ight) ight)] written in

Consider the oscillating Paul trap potential

\[U(z, ho)=\frac{U_{0}+U_{1} \cos \Omega t}{ho_{0}^{2}+2 z_{0}^{2}}\left(2 z^{2}+\left(ho_{0}^{2}-ho^{2}\right)\right)\]
written in cylindrical coordinates.

(a) Show that this potential satisfies Laplace's equation.

(b) Consider a point particle of charge \(Q\) in this potential. Analyze the dynamics using a Lagrangian and show that the particle is trapped.