Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S7 / 1

In cylindrical coordinates the orthogonal velocity components are $\dot{r}$, $r\dot{\theta }$, $\dot{z}$. Their squared magnitudes add, and $\dot{\mathbf{r}}\cdot \mathbf{A}=\dot{z}A(r)$. Thus $\boxed{\mathcal{L}=m({\dot{r}}^2+r^2{\dot{\theta }}^2+{\dot{z}}^2)/2+q\dot{z}A(r)}$. Taking the curl gives $B_{\theta }=-A^{\prime }(r)$, consistent with the prescribed field.

Velocity differentiation gives $\boxed{p_r=m\dot{r},p_{\theta }=m{r}^2\dot{\theta },p_z=m\dot{z}+qA(r)}$. The coordinates $\theta$ and $z$ are cyclic, so $p_{\theta },p_z$ are conserved. The radial momentum generally changes. In particular the conserved axial canonical momentum differs from the axial mechanical momentum by $qA$.

The Legendre transform cancels the velocity-linear magnetic term. Expressing velocities in terms of momenta givesThe Hamiltonian is time independent and hence conserved. It equals kinetic energy because a static magnetic field does no work; its dependence on $A$ is through the mechanical momentum combination.

The initial condition makes $p_{\theta }=0$ at all times. Accessible radii obey $|p_z-qA(r)|\le \sqrt{2mH}$. For $q>0$ and $p_z>\sqrt{2mH}$, monotonicity of $A$ gives exactly the two stated turning equations and confinement to the interval between their unique roots.

The squared condition alone omits a sign qualification. More generally both positive roots require $p_z/q>0$, and are the sorted roots $A(r)=(p_z\pm \sqrt{2mH})/q$. For example $q=-1$, $A=r$, $p_z=-3$, $2mH=1$ confines the orbit to $2\le r\le 4$, but the source's named $r_1,r_2$ equations reverse those labels. This is a convention issue, not a failure of confinement.

Expand $A(r)=A(r_0)+A^{\prime }(r_0)s+O(s^2)$ about $qA(r_0)=p_z$. Then $p_z-qA\simeq -q{A}^{\prime }(r_0)s$ and conservation of energy givesHere $\Omega =q{A}^{\prime }(r_0)/m$ can be signed, but the physical oscillation frequency and amplitude are nonnegative. The small-energy condition ensures that this local expansion samples only a narrow range about $r_0$, provided the field varies smoothly there.