1. The problem statement, all variables and given/known data Consider a charged particle, of mass m and charge q, confined in a device called a Penning Trap. In this device, there is a quadrupole electric field described in cartesian coordinates by the potential Phi[x,y,z] = U0 (2z^2 - x^2 - y^2) / (r0^2 + 2z0^2) Where U0 is the constant electrode potential, r0 and z0 are the radial and vertical extension of the device. There is also an uniform magnetic field through OZ of intensity B0. Determine the equations of motion for the particle and represent the respective trajectory, using Mathematica. 2. Relevant equations Lorentz Force http://gabrielse.physics.harvard.edu/gabrielse/papers/1990/1990_tjoelker/chapter_2.pdf 3. The attempt at a solution So, basically what I did was convert the potential to cylindrical coordinates and write it as such: Phi[r, θ, z] = U0 (2z^2 - r^2)/4d^2, where d 4d^2 is the denominator in the cartesian potential. Then I split the movement into radial (r) and axial (z). For axial movement, there'd be no influence from the magnetic field (it's parallel to it), so using the Lorentz force we get a simple harmonic motion: For the radial component, there IS influence from the magnetic field. Lorentz force (the cross product simplifies since the radial component is perpendicular to the field) and simplification into two frequencies (q U0 / (md^2) for wz and q B0 / (mc) for wc: But now I'm not sure what boundary conditions to apply to simplify the problem. The next step is graphing a trajectory of a particle in this Penning trap but I'm a bit lost on what to do next. Any direction is appreciated. Thank you!