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 / d / Solution

The Legendre transform cancels the term linear in velocity and givesIt equals the mechanical kinetic energy. The Hamiltonian has no explicit time dependence, so $H$ is conserved. This is also a direct consequence of the magnetic force doing no work: $q\mathbf{v}\cdot (\mathbf{v}\times \mathbf{B})=0$.