We jump straight into finding the equations of the fields involved — the magnetic field is simply , while the -field can be found via the potential :

Now, taking the Lorentz force to be the only force acting on this system, Newton's law of motion states

Essentially, what we now have is a couple of differential equations. Notice that only appears in the component, thus we shall examine that first.

The Vertical Component

Immediately we notice that the component is just a simple harmonic motion equation

With the known solution in terms of initial conditions and

The Azimuthal Components

From the and components, we have two coupled differential equations

Note that for each component, the reality of the situation means that all the values above has to be real. With that in mind, consider the equation

Owing to the linearity of the differential equation, we can define a such that , and

Using the ansatz , the above differential equation can now be rewritten,

Solving for ,

Let us first consider the case where is real. The general solution is therefore

By setting , and in terms of the initial conditions are

Here I utilise the sympy package in Python to solve for the coefficients:

>>> from sympy import *
>>> var('w_0 alpha beta omega Omega')
(w_0, alpha, beta, omega, Omega)
>>> wp = Symbol("w_0\'")
>>> coef = solve([
...         w_0-alpha-beta,
...         wp+I*((omega+Omega)*alpha + (omega-Omega)*beta)
...        ], [
...         alpha, beta
...        ])
>>> print(latex(coef))

Placing the coefficients back into the equation,

For the ease of calculation, and in anticipation for purposes of coding, we form the following variables:

Which means

Non-Oscillatory Cases

There is a possibility that , which means that . If we use the trigonometry/hyperbolic identities

Then Equation becomes

What we notice here, and also with Equation , is that in the case that either or is imaginary, then we merely take its absolute value and change the trigonometric function into its corresponding hyperbolic one (not generally true, but it is in the case of , , and ).

The Forces

From Equation , we find that the forces are given by

which we can find now that we have the solved equations of motion.