= Solution
The <energy> function $E=\sum_i\dot q_i\partial L/\partial\dot q_i-L$ is
$$
\boxed{E=\frac m2(\dot r^2+r^2\dot\theta^2+\dot z^2)+e\lambda z^2.}
$$
The velocity-linear magnetic term cancels. Since $L$ has no explicit time dependence, $dE/dt=-\partial_tL=0$. Equivalently, magnetic <force> does no <work>, and electric <work> is exactly minus the change of $e\lambda z^2$. <Energy> here is defined in the supplied time-independent gauge and equals physical kinetic plus <electrostatic potential> <energy>.
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