Exploring Quantum Physics - Week 2 Question 10

Question:

Say we have the action $S = \int{dt[\frac{1}{2}mv^2 + qEx]}$. What equation of motion does the principle of least action give?

Solution:

To minimize the classical action functional - we will use the Euler-Lagrange equations as follow:

$\begin{eqnarray*} \frac{\partial}{\partial x} L - \frac{d}{dx}\frac{\partial}{\partial v} L & = & 0 \\ qE - \frac{d}{dx} mv & = & 0 \\ ma & = & qE \\ a & = & \frac{qE}{m} \end{eqnarray*}$

There you go!