## Taming Ions: the Paul Trap

In this problem we want to understand the mechanism behind **ion trapping**. We will study the movement of charged particles in a so-called **Paul trap** by solving the equations of motion using approximative methods.

In the following problems we will be concerned with the movement of charged particles under the influence of an electrostatic field. Depending on the question we will have to solve **Euler-Lagrange equations** of motion or use similar approaches. A lot of the problems are historically motivated to which we shall refer in the respective background sections.

In this problem we want to understand the mechanism behind **ion trapping**. We will study the movement of charged particles in a so-called **Paul trap** by solving the equations of motion using approximative methods.

The **theoretical description** of the movement of a particle in some curved 3D-space and the movement of it on a 2D-surface are equivalent. Both entities are described by a **metric** and the equation of motion is the **geodesic equation**. Further giving the particle a charge lets it further interact with an **electric field**. In this problem you can learn to **derive** the corresponding geodesic equation.

A lot of **molecules** can be seen in a first approximation as **two charges with a fixed separation**. In this problem you will learn what happens if such a molecule is exposed to a **constant electric field**, especially, what kind of interesting **movements** one can observe. You will also be able to apply some of your knowledge of **mechanics**.