## The Movement of a Dipolar Molecule in a Constant Electric Field

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.

## Problem Statement

We want to assume that both charges are equal (no net charge). Furthermore they shall be separated by a fixed distance $$R$$ and have two not necessarily equal masses $$m_1$$ and $$m_2$$. Now there is a constant electrical field which may be parallel to the $$z$$-axis.

What happens to the molecule in the field? Describe its movement qualitatively!

You may want to proceed as follows:

1. find the potential energy of the molecule in the electric field,
2. determine conserved quantities using the Lagrange function and
3. discuss the movement in terms of this conserved quantity.

## Background: Symmetries and Conserved Quantities

The Lagrange function $$L\left(q_k,\dot{q}_k,t\right)$$ is given in terms of generalized coordinates $$q_k$$ and velocities $$\dot{q}_k$$. Conserved quantities are related to symmetries of the Lagrange function. If for some $$l$$$\begin{eqnarray}\frac{\partial L\left(q_{k},\dot{q}_{k},t\right)}{\partial q_{l}}&=&0 \ , \mathrm{then}\quad p_l \equiv \frac{\partial L\left(q_{k},\dot{q}_{k},t\right)}{\partial\dot{q}_{l}}\end{eqnarray}$is a conserved quantity. Do you remember why, regarding the Euler-Lagrange equation?