Trapping charged particles.

We jump straight into finding the equations of the fields involved — the magnetic field is simply

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

Immediately we notice that the

With the known solution in terms of initial conditions

From the

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

Using the ansatz

Solving for

Let us first consider the case where

By setting

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

There is a possibility that

Then Equation

What we notice here, and also with Equation

From Equation

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