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!
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!
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