Past exam of the theoretical physics I course of the University of Cambridge / 2012 / Paper 1 / 3 / b / Solution

The energy function $E={\sum }_i{\dot{q}}_i\partial L/\partial {\dot{q}}_i-L$ isThe velocity-linear magnetic term cancels. Since $L$ has no explicit time dependence, $dE/dt=-{\partial }_{t}L=0$. 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.