Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S7 / 1

Hamilton's principle states that the physical path makes the action $S={\int }_{t_i}^{t_f}L(\mathbf{q},\dot{\mathbf{q}},t)dt$ stationary under differentiable variations $\delta q_i$ vanishing at the fixed endpoints. Its conventional “least action” name does not guarantee a minimum. Expanding to first order and integrating the velocity term by parts givesThe endpoint term vanishes. By the fundamental lemma of the calculus of variations, the coefficients of the independent interior variations must vanish, yieldingFor example, the zero path of a harmonic oscillator with endpoints zero can have both positive and negative second variations when the elapsed time exceeds a half-period; stationary action then need not be a least value. The Euler-Lagrange condition is the local statement that the variational principle actually supplies.

The Euler-Lagrange equations in cylindrical coordinates areThe last two follow because $\varphi$ and $z$ are cyclic coordinates. They give the conserved quantitiesSince the Lagrangian is autonomous, its energy function is conserved as well:The velocity-linear term cancels from this energy, but it still changes the equations of motion. A dimensionless logarithm means $\log (R/R_0)$ for some fixed reference length; replacing the printed $\log R$ by that expression changes the Lagrangian only by a total time derivative and shifts the numerical value of $k$, not the trajectory.

Write $w=a^2{\Omega }^2/C$, requiring $C\ne 0$ and $a>0$. The initial conditions fix $\ell =a^2\Omega$ and $\dot{z}=w+C\log (R/a)$. At $R=a$, the radial force is $a{\Omega }^2-Cw/a=0$. Thus the unperturbed motion is the helixTo analyse a small initial radial velocity, eliminate the cyclic velocities using the conserved quantities. The radial energy equation is ${\dot{R}}^2/2+U(R)=\mathcal{E}$, withHere $U^{\prime }(a)=0$ and $U^{\prime \prime }(a)=2{\Omega }^2+C^2/a^2>0$. The circular cylindrical radius is therefore stable. For $\eta =R-a$, the first-order equation is $\ddot{\eta }+{\omega }_r^2\eta =0$, soThe orbit acquires small radial oscillations. The other coordinates are modulated consistently with the same constants:These expressions retain the stated initial angular and axial velocities. The perturbation adds ${\delta }^2/2$ to the conserved energy. The expansion assumes $|\delta |/(a{\omega }_r)\ll 1$, which follows from the supplied smallness condition when $\Omega \ne 0$; it is not an exact finite-amplitude helix.

It can describe the motion per unit mass of a charged particle in the azimuthal magnetic field of a long straight current-carrying wire. Choose magnetic vector potentialThen $\mathbf{B}=\nabla \times \mathbf{A}=(mC/(qR))\hat{\mathbf{\varphi }}$ and $(T+q\mathbf{A}\cdot \mathbf{v})/m$ is the given Lagrangian. In terms of a straight-wire current $I$, $B_{\varphi }={\mu }_{0}I/(2\pi R)$, so $C=q{\mu }_{0}I/(2\pi m)$ with signs fixed by charge and current direction.

A static magnetic force does no work, consistent with the conserved kinetic energy. Its radial component is $-q\dot{z}{B}_{\varphi }$ and its axial component is $q\dot{R}{B}_{\varphi }$, reproducing the radial and axial couplings above. The reference length in the logarithm is a gauge convention, not a new physical wire radius in the ideal exterior-field model.