Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S7 / 1 / c / Solution

Define $a=q{B}_0/(2L)>0$. Differentiating the Lagrangian with respect to each generalized velocity givesThe coordinates $y$ and $z$ are cyclic, so $p_y$ and $p_z$ are conserved. The momentum $p_x$ is generally not conserved: its equation is ${\dot{p}}_x=-2ax\dot{z}$. In particular, conserved canonical $p_z$ is not the same as conserved mechanical $m\dot{z}$, which changes as the particle moves across the magnetic-field gradient.