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

In Cartesian components, $\partial L/\partial v_i=m{v}_i+e{A}_i$ and $\partial L/\partial x_i=e{v}_j{\partial }_i{A}_j-e{\partial }_i\varphi$. The Euler-Lagrange equation therefore givesThus the minimally coupled particle Lagrangian yields the Lorentz force.

For an arbitrary smooth gauge function $\chi$, take $\mathbf{A}\to \mathbf{A}+\nabla \chi$ and $\varphi \to \varphi -{\partial }_t\chi$. Both fields are unchanged, whileThe action changes only by an endpoint term, so fixed-endpoint variation gives the same motion. Canonical momentum shifts by $e\nabla \chi$, whereas kinetic momentum $m\mathbf{v}$ and the Lorentz force are gauge invariant. This is the relevant gauge invariance of the classical particle dynamics.

The cylindrical entries of $\mathbf{A}$ are physical vector components, so $\mathbf{v}\cdot \mathbf{A}=\mu r^2\dot{\theta }$, not $\mu r\dot{\theta }$. The Lagrangian isIts three equations areFor $r>0$, the angular equation can be expanded as $r\ddot{\theta }+2\dot{r}\dot{\theta }+(2e\mu /m)\dot{r}=0$. Directly computing the fields gives $\mathbf{B}=2\mu \hat{\mathbf{z}}$ and $\mathbf{E}=-2\lambda z\hat{\mathbf{z}}$, agreeing with the equations above.

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.

The coordinate $\theta$ is cyclic. Its conjugate momentum is therefore a second conserved quantity,It differs from mechanical angular momentum $m{r}^2\dot{\theta }$. The magnetic contribution is essential to the rotational Noether charge in this axisymmetric gauge.

For constant $r=R>0$, the radial equation requires $\dot{\theta }(\dot{\theta }+2e\mu /m)=0$. Select the stipulated nonzero angular velocity and define the positive cyclotron frequency ${\Omega }_c=2e\mu /m$. ThenThis is a circle in the transverse plane combined with an independent axial oscillation, as in uniform magnetic field with axial harmonic confinement. Viewed from positive $z$, positive charge rotates clockwise. Its three-dimensional path lies on the cylinder $r=R$ and reverses its vertical motion; it is generally not a constant-pitch helix. For $Z=0$ it is a horizontal circle. This orbit has $p_{\theta }=-e\mu R^2$.

For the specified parameter relation,Thus there are $|n|$ axial oscillations per transverse revolution, and the nondegenerate three-dimensional trajectory closes after $2\pi /{\Omega }_c$. Since $\lambda >0$, the integer $n=0$ is excluded; changing its sign changes nothing. More generally commensurate cyclotron and axial oscillations close for any rational frequency ratio, whereas irrational ratios give quasiperiodic motion when both components are present. An integer ratio is sufficient, not the only closure condition. The same frequency argument also applies to transverse cyclotron circles centred away from the chosen cylindrical origin.