# Constants of motion of an electron in a harmonic electromagnetic field in free space

I have encountered a question in Classical Electrodynamics, as below:

In free space, an electron, initially at rest at $$z=0$$, is subjected to an intense laser field $$\vec E=\hat x A \cos(\omega t-\frac{\omega z}{c})$$, and $$\vec B=\hat y \frac{A}{c} \cos(\omega t-\frac{\omega z}{c})$$. Show that $$\gamma - \frac{p_z}{mc}$$ is a constant of motion.

To answer this, I solved for the relativistic Lagrangian using $$L=-\frac{mc^2}{\gamma}-e\phi+e\vec{A}\cdot \vec{v}$$ $$\Rightarrow L=-\frac{mc^2}{\gamma}+eA \cos( \omega t -\frac{ \omega z}{c}) \left[x+\frac{1}{2c}(zv_x-xv_z)\right]$$ But this L does not yield any constants of motion as in the question.

Can someone tell me where I am going wrong?

• Why do you expect to eyeball it from the Lagrangian? The question is posed as initial value problem. You should begin with equations of motion propagating the initial condition and check if the evolution is consistent with such constant of motion. – Ján Lalinský Apr 12 at 21:44