 Solution

ID: 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 gives

H = \frac{p _{x}^{2} + p _{y}^{2} + ( p _{z} + a x ^{2} ) ^{2}}{2 m} .

It 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 v \cdot (v \times B) = 0

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