# Part II, 2015

### Jump to course

Paper 1, Section II, F

commentLet $k$ be an algebraically closed field.

(i) Let $X$ and $Y$ be affine varieties defined over $k$. Given a map $f: X \rightarrow Y$, define what it means for $f$ to be a morphism of affine varieties.

(ii) With $X, Y$ still affine varieties over $k$, show that there is a one-to-one correspondence between $\operatorname{Hom}(X, Y)$, the set of morphisms between $X$ and $Y$, and $\operatorname{Hom}(A(Y), A(X))$, the set of $k$-algebra homomorphisms between $A(Y)$ and $A(X)$.

(iii) Let $f: \mathbb{A}^{2} \rightarrow \mathbb{A}^{4}$ be given by $f(t, u)=\left(u, t, t^{2}, t u\right)$. Show that the image of $f$ is an affine variety $X$, and find a set of generators for $I(X)$.

Paper 2, Section II, F

comment(i) Define the radical of an ideal.

(ii) Assume the following statement: If $k$ is an algebraically closed field and $I \subseteq$ $k\left[x_{1}, \ldots, x_{n}\right]$ is an ideal, then either $I=(1)$ or $Z(I) \neq \emptyset$. Prove the Hilbert Nullstellensatz, namely that if $I \subseteq k\left[x_{1}, \ldots, x_{n}\right]$ with $k$ algebraically closed, then

$I(Z(I))=\sqrt{I}$

(iii) Show that if $A$ is a commutative ring and $I, J \subseteq A$ are ideals, then

$\sqrt{I \cap J}=\sqrt{I} \cap \sqrt{J} .$

(iv) Is

$\sqrt{I+J}=\sqrt{I}+\sqrt{J} ?$

Give a proof or a counterexample.

Paper 3, Section II, F

comment(i) Let $X$ be an affine variety. Define the tangent space of $X$ at a point $P$. Say what it means for the variety to be singular at $P$.

(ii) Find the singularities of the surface in $\mathbb{P}^{3}$ given by the equation

$x y z+y z w+z w x+w x y=0 .$

(iii) Consider $C=Z\left(x^{2}-y^{3}\right) \subseteq \mathbb{A}^{2}$. Let $X \rightarrow \mathbb{A}^{2}$ be the blowup of the origin. Compute the proper transform of $C$ in $X$, and show it is non-singular.

Paper 4, Section II, F

comment(i) Explain how a linear system on a curve $C$ may induce a morphism from $C$ to projective space. What condition on the linear system is necessary to yield a morphism $f: C \rightarrow \mathbb{P}^{n}$ such that the pull-back of a hyperplane section is an element of the linear system? What condition is necessary to imply the morphism is an embedding?

(ii) State the Riemann-Roch theorem for curves.

(iii) Show that any divisor of degree 5 on a curve $C$ of genus 2 induces an embedding.

Paper 1, Section II, H

commentState carefully a version of the Seifert-van Kampen theorem for a cover of a space by two closed sets.

Let $X$ be the space obtained by gluing together a Möbius band $M$ and a torus $T=S^{1} \times S^{1}$ along a homeomorphism of the boundary of $M$ with $S^{1} \times\{1\} \subset T$. Find a presentation for the fundamental group of $X$, and hence show that it is infinite and non-abelian.

Paper 2, Section II, H

commentDefine what it means for $p: \widetilde{X} \rightarrow X$ to be a covering map, and what it means to say that $p$ is a universal cover.

Let $p: \tilde{X} \rightarrow X$ be a universal cover, $A \subset X$ be a locally path connected subspace, and $\tilde{A} \subset p^{-1}(A)$ be a path component containing a point $\tilde{a}_{0}$ with $p\left(\tilde{a}_{0}\right)=a_{0}$. Show that the restriction $\left.p\right|_{\tilde{A}}: \widetilde{A} \rightarrow A$ is a covering map, and that under the Galois correspondence it corresponds to the subgroup

$\operatorname{Ker}\left(\pi_{1}\left(A, a_{0}\right) \rightarrow \pi_{1}\left(X, a_{0}\right)\right)$

of $\pi_{1}\left(A, a_{0}\right)$.

Paper 3, Section II, H

commentLet $K$ and $L$ be simplicial complexes. Explain what is meant by a simplicial approximation to a continuous map $f:|K| \rightarrow|L|$. State the simplicial approximation theorem, and define the homomorphism induced on homology by a continuous map between triangulable spaces. [You do not need to show that the homomorphism is welldefined.]

Let $h: S^{1} \rightarrow S^{1}$ be given by $z \mapsto z^{n}$ for a positive integer $n$, where $S^{1}$ is considered as the unit complex numbers. Compute the map induced by $h$ on homology.

Paper 4, Section II, H

commentState the Mayer-Vietoris theorem for a simplicial complex $K$ which is the union of two subcomplexes $M$ and $N$. Explain briefly how the connecting homomorphism $\partial_{n}: H_{n}(K) \rightarrow H_{n-1}(M \cap N)$ is defined.

If $K$ is the union of subcomplexes $M_{1}, M_{2}, \ldots, M_{n}$, with $n \geqslant 2$, such that each intersection

$M_{i_{1}} \cap M_{i_{2}} \cap \cdots \cap M_{i_{k}}, \quad 1 \leqslant k \leqslant n,$

is either empty or has the homology of a point, then show that

$H_{i}(K)=0 \quad \text { for } \quad i \geqslant n-1 .$

Construct examples for each $n \geqslant 2$ showing that this is sharp.

Paper 1, Section II, A

commentDefine the Rayleigh-Ritz quotient $R[\psi]$ for a normalisable state $|\psi\rangle$ of a quantum system with Hamiltonian $H$. Given that the spectrum of $H$ is discrete and that there is a unique ground state of energy $E_{0}$, show that $R[\psi] \geqslant E_{0}$ and that equality holds if and only if $|\psi\rangle$ is the ground state.

A simple harmonic oscillator (SHO) is a particle of mass $m$ moving in one dimension subject to the potential

$V(x)=\frac{1}{2} m \omega^{2} x^{2}$

Estimate the ground state energy $E_{0}$ of the SHO by using the ground state wavefunction for a particle in an infinite potential well of width $a$, centred on the origin (the potential is $U(x)=0$ for $|x|<a / 2$ and $U(x)=\infty$ for $|x|>a / 2)$. Take $a$ as the variational parameter.

Perform a similar estimate for the energy $E_{1}$ of the first excited state of the SHO by using the first excited state of the infinite potential well as a trial wavefunction.

Is the estimate for $E_{1}$ necessarily an upper bound? Justify your answer.

$\left[\right.$ You may use : $\int_{-\pi / 2}^{\pi / 2} y^{2} \cos ^{2} y d y=\frac{\pi}{4}\left(\frac{\pi^{2}}{6}-1\right) \quad$ and $\left.\quad \int_{-\pi}^{\pi} y^{2} \sin ^{2} y d y=\pi\left(\frac{\pi^{2}}{3}-\frac{1}{2}\right) \cdot\right]$

Paper 2, Section II, A

commentA beam of particles of mass $m$ and energy $\hbar^{2} k^{2} / 2 m$ is incident on a target at the origin described by a spherically symmetric potential $V(r)$. Assuming the potential decays rapidly as $r \rightarrow \infty$, write down the asymptotic form of the wavefunction, defining the scattering amplitude $f(\theta)$.

Consider a free particle with energy $\hbar^{2} k^{2} / 2 m$. State, without proof, the general axisymmetric solution of the Schrödinger equation for $r>0$ in terms of spherical Bessel and Neumann functions $j_{\ell}$ and $n_{\ell}$, and Legendre polynomials $P_{\ell}(\ell=0,1,2, \ldots)$. Hence define the partial wave phase shifts $\delta_{\ell}$ for scattering from a potential $V(r)$ and derive the partial wave expansion for $f(\theta)$ in terms of phase shifts.

Now suppose

$V(r)=\left\{\begin{array}{cc} \hbar^{2} \gamma^{2} / 2 m & r<a \\ 0 & r>a \end{array}\right.$

with $\gamma>k$. Show that the S-wave phase shift $\delta_{0}$ obeys

$\frac{\tanh (\kappa a)}{\kappa a}=\frac{\tan \left(k a+\delta_{0}\right)}{k a}$

where $\kappa^{2}=\gamma^{2}-k^{2}$. Deduce that for an S-wave solution

$f \rightarrow \frac{\tanh \gamma a-\gamma a}{\gamma} \quad \text { as } \quad k \rightarrow 0$

[You may assume : $\quad \exp (i k r \cos \theta)=\sum_{\ell=0}^{\infty}(2 \ell+1) i^{\ell} j_{\ell}(k r) P_{\ell}(\cos \theta)$

and $j_{\ell}(\rho) \sim \frac{1}{\rho} \sin (\rho-\ell \pi / 2), \quad n_{\ell}(\rho) \sim-\frac{1}{\rho} \cos (\rho-\ell \pi / 2) \quad$ as $\left.\quad \rho \rightarrow \infty .\right]$

Paper 3, Section II, A

commentA particle of mass $m$ and energy $E=-\hbar^{2} \kappa^{2} / 2 m<0$ moves in one dimension subject to a periodic potential

$V(x)=-\frac{\hbar^{2} \lambda}{m} \sum_{\ell=-\infty}^{\infty} \delta(x-\ell a) \quad \text { with } \quad \lambda>0 .$

Determine the corresponding Floquet matrix $\mathcal{M}$. [You may assume without proof that for the Schrödinger equation with potential $\alpha \delta(x)$ the wavefunction $\psi(x)$ is continuous at $x=0$ and satisfies $\left.\psi^{\prime}(0+)-\psi^{\prime}(0-)=\left(2 m \alpha / \hbar^{2}\right) \psi(0) .\right]$

Explain briefly, with reference to Bloch's theorem, how restrictions on the energy of a Bloch state can be derived from $\mathcal{M}$. Deduce that for the potential $V(x)$ above, $\kappa$ is confined to a range whose boundary values are determined by

$\tanh \left(\frac{\kappa a}{2}\right)=\frac{\kappa}{\lambda} \quad \text { and } \quad \operatorname{coth}\left(\frac{\kappa a}{2}\right)=\frac{\kappa}{\lambda} .$

Sketch the left-hand and right-hand sides of each of these equations as functions of $y=\kappa a / 2$. Hence show that there is exactly one allowed band of negative energies with either (i) $E_{-} \leqslant E<0$ or (ii) $E_{-} \leqslant E \leqslant E_{+}<0$ and determine the values of $\lambda a$ for which each of these cases arise. [You should not attempt to evaluate the constants $E_{\pm} .$]

Comment briefly on the limit $a \rightarrow \infty$ with $\lambda$ fixed.

Paper 4, Section II,

commentLet $\Lambda$ be a Bravais lattice with basis vectors $\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3}$. Define the reciprocal lattice $\Lambda^{*}$ and write down basis vectors $\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}$ for $\Lambda^{*}$ in terms of the basis for $\Lambda$.

A finite crystal consists of identical atoms at sites of $\Lambda$ given by

$\ell=n_{1} \mathbf{a}_{1}+n_{2} \mathbf{a}_{2}+n_{3} \mathbf{a}_{3} \quad \text { with } \quad 0 \leqslant n_{i}<N_{i}$

A particle of mass $m$ scatters off the crystal; its wavevector is $\mathbf{k}$ before scattering and $\mathbf{k}^{\prime}$ after scattering, with $|\mathbf{k}|=\left|\mathbf{k}^{\prime}\right|$. Show that the scattering amplitude in the Born approximation has the form

$-\frac{m}{2 \pi \hbar^{2}} \Delta(\mathbf{q}) \tilde{U}(\mathbf{q}), \quad \mathbf{q}=\mathbf{k}^{\prime}-\mathbf{k}$

where $U(\mathbf{x})$ is the potential due to a single atom at the origin and $\Delta(\mathbf{q})$ depends on the crystal structure. [You may assume that in the Born approximation the amplitude for scattering off a potential $V(\mathbf{x})$ is $-\left(m / 2 \pi \hbar^{2}\right) \tilde{V}(\mathbf{q})$ where tilde denotes the Fourier transform.]

Derive an expression for $|\Delta(\mathbf{q})|$ that is valid when $e^{-i \mathbf{q} \cdot \mathbf{a}_{i}} \neq 1$. Show also that when $\mathbf{q}$ is a reciprocal lattice vector $|\Delta(\mathbf{q})|$ is equal to the total number of atoms in the crystal. Comment briefly on the significance of these results.

Now suppose that $\Lambda$ is a face-centred-cubic lattice:

$\mathbf{a}_{1}=\frac{a}{2}(\hat{\mathbf{y}}+\hat{\mathbf{z}}), \quad \mathbf{a}_{2}=\frac{a}{2}(\hat{\mathbf{z}}+\hat{\mathbf{x}}), \quad \mathbf{a}_{3}=\frac{a}{2}(\hat{\mathbf{x}}+\hat{\mathbf{y}})$

where $a$ is a constant. Show that for a particle incident with $|\mathbf{k}|>2 \pi / a$, enhanced scattering is possible for at least two values of the scattering angle, $\theta_{1}$ and $\theta_{2}$, related by

$\frac{\sin \left(\theta_{1} / 2\right)}{\sin \left(\theta_{2} / 2\right)}=\frac{\sqrt{3}}{2}$

Paper 1, Section II, K

comment(a) Give the definition of a birth and death chain in terms of its generator. Show that a measure $\pi$ is invariant for a birth and death chain if and only if it solves the detailed balance equations.

(b) There are $s$ servers in a post office and a single queue. Customers arrive as a Poisson process of rate $\lambda$ and the service times at each server are independent and exponentially distributed with parameter $\mu$. Let $X_{t}$ denote the number of customers in the post office at time $t$. Find conditions on $\lambda, \mu$ and $s$ for $X$ to be positive recurrent, null recurrent and transient, justifying your answers.

Paper 2, Section II, $24 K$

comment(i) Defne a Poisson process on $\mathbb{R}_{+}$with rate $\lambda$. Let $N$ and $M$ be two independent Poisson processes on $\mathbb{R}_{+}$of rates $\lambda$ and $\mu$ respectively. Prove that $N+M$ is also a Poisson process and find its rate.

(ii) Let $X$ be a discrete time Markov chain with transition matrix $K$ on the finite state space $S$. Find the generator of the continuous time Markov chain $Y_{t}=X_{N_{t}}$ in terms of $K$ and $\lambda$. Show that if $\pi$ is an invariant distribution for $X$, then it is also invariant for $Y$.

Suppose that $X$ has an absorbing state $a$. If $\tau_{a}$ and $T_{a}$ are the absorption times for $X$ and $Y$ respectively, write an equation that relates $\mathbb{E}_{x}\left[\tau_{a}\right]$ and $\mathbb{E}_{x}\left[T_{a}\right]$, where $x \in S$.

[Hint: You may want to prove that if $\xi_{1}, \xi_{2}, \ldots$ are i.i.d. non-negative random variables with $\mathbb{E}\left[\xi_{1}\right]<\infty$ and $M$ is an independent non-negative random variable, then $\left.\mathbb{E}\left[\sum_{i=1}^{M} \xi_{i}\right]=\mathbb{E}[M] \mathbb{E}\left[\xi_{1}\right] .\right]$

Paper 3, Section II, K

comment(i) Let $X$ be a Poisson process of parameter $\lambda$. Let $Y$ be obtained by taking each point of $X$ and, independently of the other points, keeping it with probability $p$. Show that $Y$ is another Poisson process and find its intensity. Show that for every fixed $t$ the random variables $Y_{t}$ and $X_{t}-Y_{t}$ are independent.

(ii) Suppose we have $n$ bins, and balls arrive according to a Poisson process of rate 1 . Upon arrival we choose a bin uniformly at random and place the ball in it. We let $M_{n}$ be the maximum number of balls in any bin at time $n$. Show that

$\mathbb{P}\left(M_{n} \geqslant(1+\epsilon) \frac{\log n}{\log \log n}\right) \rightarrow 0 \quad \text { as } n \rightarrow \infty$

[You may use the fact that if $\xi$ is a Poisson random variable of mean 1 , then

$\mathbb{P}(\xi \geqslant x) \leqslant \exp (x-x \log x) .]$

Paper 4, Section II, K

comment(i) Let $X$ be a Markov chain on $S$ and $A \subset S$. Let $T_{A}$ be the hitting time of $A$ and $\tau_{y}$ denote the total time spent at $y \in S$ by the chain before hitting $A$. Show that if $h(x)=\mathbb{P}_{x}\left(T_{A}<\infty\right)$, then $\mathbb{E}_{x}\left[\tau_{y} \mid T_{A}<\infty\right]=[h(y) / h(x)] \mathbb{E}_{x}\left(\tau_{y}\right) .$

(ii) Define the Moran model and show that if $X_{t}$ is the number of individuals carrying allele $a$ at time $t \geqslant 0$ and $\tau$ is the fixation time of allele $a$, then

$\mathbb{P}\left(X_{\tau}=N \mid X_{0}=i\right)=\frac{i}{N}$

Show that conditionally on fixation of an allele $a$ being present initially in $i$ individuals,

$\mathbb{E}[\tau \mid \text { fixation }]=N-i+\frac{N-i}{i} \sum_{j=1}^{i-1} \frac{j}{N-j}$

Paper 1, Section II, C

comment(a) State the integral expression for the gamma function $\Gamma(z)$, for $\operatorname{Re}(z)>0$, and express the integral

$\int_{0}^{\infty} t^{\gamma-1} e^{i t} d t, \quad 0<\gamma<1$

in terms of $\Gamma(\gamma)$. Explain why the constraints on $\gamma$ are necessary.

(b) Show that

$\int_{0}^{\infty} \frac{e^{-k t^{2}}}{\left(t^{2}+t\right)^{\frac{1}{4}}} d t \sim \sum_{m=0}^{\infty} \frac{a_{m}}{k^{\alpha+\beta m}}, \quad k \rightarrow \infty$

for some constants $a_{m}, \alpha$ and $\beta$. Determine the constants $\alpha$ and $\beta$, and express $a_{m}$ in terms of the gamma function.

State without proof the basic result needed for the rigorous justification of the above asymptotic formula.

[You may use the identity:

$\left.(1+z)^{\alpha}=\sum_{m=0}^{\infty} c_{m} z^{m}, \quad c_{m}=\frac{\Gamma(\alpha+1)}{m ! \Gamma(\alpha+1-m)}, \quad|z|<1 .\right]$

Paper 3, Section II, $27 \mathrm{C}$

commentShow that

$\int_{0}^{1} e^{i k t^{3}} d t=I_{1}-I_{2}, \quad k>0$

where $I_{1}$ is an integral from 0 to $\infty$ along the line $\arg (z)=\frac{\pi}{6}$ and $I_{2}$ is an integral from 1 to $\infty$ along a steepest-descent contour $C$ which you should determine.

By employing in the integrals $I_{1}$ and $I_{2}$ the changes of variables $u=-i z^{3}$ and $u=-i\left(z^{3}-1\right)$, respectively, compute the first two terms of the large $k$ asymptotic expansion of the integral above.

Paper 4, Section II, C

commentConsider the ordinary differential equation

$\frac{d^{2} u}{d z^{2}}+f(z) \frac{d u}{d z}+g(z) u=0$

where

$f(z) \sim \sum_{m=0}^{\infty} \frac{f_{m}}{z^{m}}, \quad g(z) \sim \sum_{m=0}^{\infty} \frac{g_{m}}{z^{m}}, \quad z \rightarrow \infty$

and $f_{m}, g_{m}$ are constants. Look for solutions in the asymptotic form

$u(z)=e^{\lambda z} z^{\mu}\left[1+\frac{a}{z}+\frac{b}{z^{2}}+O\left(\frac{1}{z^{3}}\right)\right], \quad z \rightarrow \infty$

and determine $\lambda$ in terms of $\left(f_{0}, g_{0}\right)$, as well as $\mu$ in terms of $\left(\lambda, f_{0}, f_{1}, g_{1}\right)$.

Deduce that the Bessel equation

$\frac{d^{2} u}{d z^{2}}+\frac{1}{z} \frac{d u}{d z}+\left(1-\frac{\nu^{2}}{z^{2}}\right) u=0$

where $\nu$ is a complex constant, has two solutions of the form

$\begin{aligned} &u^{(1)}(z)=\frac{e^{i z}}{z^{1 / 2}}\left[1+\frac{a^{(1)}}{z}+O\left(\frac{1}{z^{2}}\right)\right], \quad z \rightarrow \infty \\ &u^{(2)}(z)=\frac{e^{-i z}}{z^{1 / 2}}\left[1+\frac{a^{(2)}}{z}+O\left(\frac{1}{z^{2}}\right)\right], \quad z \rightarrow \infty \end{aligned}$

and determine $a^{(1)}$ and $a^{(2)}$ in terms of $\nu .$

Can the above asymptotic expansions be valid for all $\arg (z)$, or are they valid only in certain domains of the complex $z$-plane? Justify your answer briefly.

Paper 1, Section I, D

comment(a) The action for a one-dimensional dynamical system with a generalized coordinate $q$ and Lagrangian $L$ is given by

$S=\int_{t_{1}}^{t_{2}} L(q, \dot{q}, t) d t$

State the principle of least action and derive the Euler-Lagrange equation.

(b) A planar spring-pendulum consists of a light rod of length $l$ and a bead of mass $m$, which is able to slide along the rod without friction and is attached to the ends of the rod by two identical springs of force constant $k$ as shown in the figure. The rod is pivoted at one end and is free to swing in a vertical plane under the influence of gravity.

(i) Identify suitable generalized coordinates and write down the Lagrangian of the system.

(ii) Derive the equations of motion.

Paper 2, Section I, D

commentThe Lagrangian for a heavy symmetric top of mass $M$, pinned at a point that is a distance $l$ from the centre of mass, is

$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-M g l \cos \theta$

(a) Find all conserved quantities. In particular, show that $\omega_{3}$, the spin of the top, is constant.

(b) Show that $\theta$ obeys the equation of motion

$I_{1} \ddot{\theta}=-\frac{d V_{\text {eff }}}{d \theta},$

where the explicit form of $V_{\text {eff }}$ should be determined.

(c) Determine the condition for uniform precession with no nutation, that is $\dot{\theta}=0$ and $\dot{\phi}=$ const. For what values of $\omega_{3}$ does such uniform precession occur?

Paper 2, Section II, C

comment(a) Consider a Lagrangian dynamical system with one degree of freedom. Write down the expression for the Hamiltonian of the system in terms of the generalized velocity $\dot{q}$, momentum $p$, and the Lagrangian $L(q, \dot{q}, t)$. By considering the differential of the Hamiltonian, or otherwise, derive Hamilton's equations.

Show that if $q$ is ignorable (cyclic) with respect to the Lagrangian, i.e. $\partial L / \partial q=0$, then it is also ignorable with respect to the Hamiltonian.

(b) A particle of charge $q$ and mass $m$ moves in the presence of electric and magnetic fields such that the scalar and vector potentials are $\phi=y E$ and $\mathbf{A}=(0, x B, 0)$, where $(x, y, z)$ are Cartesian coordinates and $E, B$ are constants. The Lagrangian of the particle is

$L=\frac{1}{2} m \dot{\mathbf{r}}^{2}-q \phi+q \dot{\mathbf{r}} \cdot \mathbf{A}$

Starting with the Lagrangian, derive an explicit expression for the Hamiltonian and use Hamilton's equations to determine the motion of the particle.

Paper 3, Section I, $7 \mathrm{D}$

comment(a) Consider a particle of mass $m$ that undergoes periodic motion in a one-dimensional potential $V(q)$. Write down the Hamiltonian $H(p, q)$ for the system. Explain what is meant by the angle-action variables $(\theta, I)$ of the system and write down the integral expression for the action variable $I$.

(b) For $V(q)=\frac{1}{2} m \omega^{2} q^{2}$ and fixed total energy $E$, describe the shape of the trajectories in phase-space. By using the expression for the area enclosed by the trajectory, or otherwise, find the action variable $I$ in terms of $\omega$ and $E$. Hence describe how $E$ changes with $\omega$ if $\omega$ varies slowly with time. Justify your answer.

Paper 4, Section I, D

commentA triatomic molecule is modelled by three masses moving in a line while connected to each other by two identical springs of force constant $k$ as shown in the figure.

(a) Write down the Lagrangian and derive the equations describing the motion of the atoms.

(b) Find the normal modes and their frequencies. What motion does the lowest frequency represent?

Paper 4, Section II, C

commentConsider a rigid body with angular velocity $\boldsymbol{\omega}$, angular momentum $\mathbf{L}$ and position vector $\mathbf{r}$, in its body frame.

(a) Use the expression for the kinetic energy of the body,

$\frac{1}{2} \int d^{3} \mathbf{r} \rho(\mathbf{r}) \dot{\mathbf{r}}^{2},$

to derive an expression for the tensor of inertia of the body, I. Write down the relationship between $\mathbf{L}, \mathbf{I}$ and $\boldsymbol{\omega}$.

(b) Euler's equations of torque-free motion of a rigid body are

$\begin{aligned} &I_{1} \dot{\omega}_{1}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3} \\ &I_{2} \dot{\omega}_{2}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \\ &I_{3} \dot{\omega}_{3}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2} \end{aligned}$

Working in the frame of the principal axes of inertia, use Euler's equations to show that the energy $E$ and the squared angular momentum $\mathbf{L}^{2}$ are conserved.

(c) Consider a cuboid with sides $a, b$ and $c$, and with mass $M$ distributed uniformly.

(i) Use the expression for the tensor of inertia derived in (a) to calculate the principal moments of inertia of the body.

(ii) Assume $b=2 a$ and $c=4 a$, and suppose that the initial conditions are such that

$\mathbf{L}^{2}=2 I_{2} E$

with the initial angular velocity $\omega$ perpendicular to the intermediate principal axis $\mathbf{e}_{2}$. Derive the first order differential equation for $\omega_{2}$ in terms of $E, M$ and $a$ and hence determine the long-term behaviour of $\boldsymbol{\omega}$.

Paper 1, Section I, $3 G$

commentLet $\mathcal{A}$ be a finite alphabet. Explain what is meant by saying that a binary code $c: \mathcal{A} \rightarrow\{0,1\}^{*}$ has minimum distance $\delta$. If $c$ is such a binary code with minimum distance $\delta$, show that $c$ is $\delta-1$ error-detecting and $\left\lfloor\frac{1}{2}(\delta-1)\right\rfloor$ error-correcting.

Show that it is possible to construct a code that has minimum distance $\delta$ for any integer $\delta>0$.

Paper 1, Section II, G

commentDefine the Hamming code. Show that it is a perfect, linear, 1-error correcting code.

I wish to send a message through a noisy channel to a friend. The message consists of a large number $N=1,000$ of letters from a 16 -letter alphabet $\mathcal{A}$. When my friend has decoded the message, she can tell whether there have been any errors. If there have, she asks me to send the message again and this is repeated until she has received the message without error. For each individual binary digit that is transmitted, there is independently a small probability $p=0.001$ of an error.

(a) Suppose that I encode my message by writing each letter as a 4-bit binary string. The whole message is then $4 N$ bits long. What is the probability $P$ that the entire message is transmitted without error? How many times should I expect to transmit the message until my friend receives it without error?

(b) As an alternative, I use the Hamming code to encode each letter of $\mathcal{A}$ as a 7-bit binary string. What is the probability that my friend can decode a single 7-bit string correctly? Deduce that the probability $Q$ that the entire message is correctly decoded is given approximately by

$Q \simeq\left(1-21 p^{2}\right)^{N} \simeq \exp \left(-21 N p^{2}\right)$

Which coding method is better?

Paper 2, Section I, G

commentA random variable $A$ takes values in the alphabet $\mathcal{A}=\{a, b, c, d, e\}$ with probabilities $0.4,0.2,0.2,0.1$ and $0.1$. Calculate the entropy of $A$.

Define what it means for a code for a general finite alphabet to be optimal. Find such a code for the distribution above and show that there are optimal codes for this distribution with differing lengths of codeword.

[You may use any results from the course without proof. Note that $\log _{2} 5 \simeq 2.32$.]

Paper 2, Section II, G

commentBriefly describe the $R S A$ public key cipher.

Just before it went into liquidation, the Internet Bank decided that it wanted to communicate with each of its customers using an RSA cipher. So, it chose a large modulus $N$, which is the product of two large prime numbers, and chose encrypting exponents $e_{j}$ and decrypting exponents $d_{j}$ for each customer $j$. The bank published $N$ and $e_{j}$ and sent the decrypting exponent $d_{j}$ secretly to customer $j$. Show explicitly that the cipher can be broken by each customer.

The bank sent out the same message to each customer. I am not a customer of the bank but have two friends who are and I notice that their published encrypting exponents are coprime. Explain how I can find the original message. Can I break the cipher?

Paper 3, Section I, G

commentLet $A$ be a random variable that takes each value $a$ in the finite alphabet $\mathcal{A}$ with probability $p(a)$. Show that, if each $l(a)$ is an integer greater than 0 and $\sum 2^{-l(a)} \leqslant 1$, then there is a decodable binary code $c: \mathcal{A} \rightarrow\{0,1\}^{*}$ with each codeword $c(a)$ having length $l(a)$.

Prove that, for any decodable code $c: \mathcal{A} \rightarrow\{0,1\}^{*}$, we have

$H(A) \leqslant \mathbb{E} l(A)$

where $H(A)$ is the entropy of the random variable $A$. When is there equality in this inequality?

Paper 4, Section I, G

commentExplain how to construct binary Reed-Muller codes. State and prove a result giving the minimum distance for each such Reed-Muller code.

Paper 1, Section I, C

commentConsider three galaxies $O, A$ and $B$ with position vectors $\mathbf{r}_{O}, \mathbf{r}_{A}$ and $\mathbf{r}_{B}$ in a homogeneous universe. Assuming they move with non-relativistic velocities $\mathbf{v}_{O}=\mathbf{0}, \mathbf{v}_{A}$ and $\mathbf{v}_{B}$, show that spatial homogeneity implies that the velocity field $\mathbf{v}(\mathbf{r})$ satisfies

$\mathbf{v}\left(\mathbf{r}_{B}-\mathbf{r}_{A}\right)=\mathbf{v}\left(\mathbf{r}_{B}-\mathbf{r}_{O}\right)-\mathbf{v}\left(\mathbf{r}_{A}-\mathbf{r}_{O}\right)$

and hence that $\mathbf{v}$ is linearly related to $\mathbf{r}$ by

$v_{i}=\sum_{j=1}^{3} H_{i j} r_{j}$

where the components of the matrix $H_{i j}$ are independent of $\mathbf{r}$.

Suppose the matrix $H_{i j}$ has the form

$H_{i j}=\frac{D}{t}\left(\begin{array}{ccc} 5 & -1 & -2 \\ 1 & 5 & -1 \\ 2 & 1 & 5 \end{array}\right)$

with $D>0$ constant. Describe the kinematics of the cosmological expansion.

Paper 1, Section II, C

commentA closed universe contains black-body radiation, has a positive cosmological constant $\Lambda$, and is governed by the equation

$\frac{\dot{a}^{2}}{a^{2}}=\frac{\Gamma}{a^{4}}-\frac{1}{a^{2}}+\frac{\Lambda}{3},$

where $a(t)$ is the scale factor and $\Gamma$ is a positive constant. Using the substitution $y=a^{2}$ and the boundary condition $y(0)=0$, deduce the boundary condition for $\dot{y}(0)$ and show that

$\ddot{y}=\frac{4 \Lambda}{3} y-2$

and hence that

$a^{2}(t)=\frac{3}{2 \Lambda}\left[1-\cosh \left(\sqrt{\frac{4 \Lambda}{3}} t\right)+\lambda \sinh \left(\sqrt{\frac{4 \Lambda}{3}} t\right)\right]$

Express the constant $\lambda$ in terms of $\Lambda$ and $\Gamma$.

Sketch the graphs of $a(t)$ for the cases $\lambda>1$ and $0<\lambda<1$.

Paper 2, Section I, C

commentThe mass density perturbation equation for non-relativistic matter $\left(P \ll \rho c^{2}\right)$ with wave number $k$ in the late universe $\left(t>t_{\mathrm{eq}}\right)$ is

$\ddot{\delta}+2 \frac{\dot{a}}{a} \dot{\delta}-\left(4 \pi G \rho-\frac{c_{s}^{2} k^{2}}{a^{2}}\right) \delta=0 .$

Suppose that a non-relativistic fluid with the equation of state $P \propto \rho^{4 / 3}$ dominates the universe when $a(t)=t^{2 / 3}$, and the curvature and the cosmological constant can be neglected. Show that the sound speed can be written in the form $c_{s}^{2}(t) \equiv d P / d \rho=$ $\bar{c}_{s}^{2} t^{-2 / 3}$ where $\bar{c}_{s}$ is a constant.

Find power-law solutions to $(*)$ of the form $\delta \propto t^{\beta}$ and hence show that the general solution is

$\delta=A_{k} t^{n_{+}}+B_{k} t^{n_{-}}$

where

$n_{\pm}=-\frac{1}{6} \pm\left[\left(\frac{5}{6}\right)^{2}-\bar{c}_{s}^{2} k^{2}\right]^{1 / 2}$

Interpret your solutions in the two regimes $k \ll k_{J}$ and $k \gg k_{J}$ where $k_{J}=\frac{5}{6 \bar{c}_{s}}$.

Paper 3, Section I, C

commentWhat is the flatness problem? Show by reference to the Friedmann equation how a period of accelerated expansion of the scale factor $a(t)$ in the early stages of the universe can solve the flatness problem if $\rho+3 P<0$, where $\rho$ is the mass density and $P$ is the pressure.

In the very early universe, where we can neglect the spatial curvature and the cosmological constant, there is a homogeneous scalar field $\phi$ with a vacuum potential energy

$V(\phi)=m^{2} \phi^{2},$

and the Friedmann energy equation (in units where $8 \pi G=1$ ) is

$3 H^{2}=\frac{1}{2} \dot{\phi}^{2}+V(\phi),$

where $H$ is the Hubble parameter. The field $\phi$ obeys the evolution equation

$\ddot{\phi}+3 H \dot{\phi}+\frac{d V}{d \phi}=0$

During inflation, $\phi$ evolves slowly after starting from a large initial value $\phi_{i}$ at $t=0$. State what is meant by the slow-roll approximation. Show that in this approximation,

$\begin{aligned} \phi(t) &=\phi_{i}-\frac{2}{\sqrt{3}} m t \\ a(t) &=a_{i} \exp \left[\frac{m \phi_{i}}{\sqrt{3}} t-\frac{1}{3} m^{2} t^{2}\right]=a_{i} \exp \left[\frac{\phi_{i}^{2}-\phi^{2}(t)}{4}\right] \end{aligned}$

where $a_{i}$ is the initial value of $a$.

As $\phi(t)$ decreases from its initial value $\phi_{i}$, what is its approximate value when the slow-roll approximation fails?

Paper 3, Section II, C

commentMassive particles and antiparticles each with mass $m$ and respective number densities $n(t)$ and $\bar{n}(t)$ are present at time $t$ in the radiation era of an expanding universe with zero curvature and no cosmological constant. Assuming they interact with crosssection $\sigma$ at speed $v$, explain, by identifying the physical significance of each of the terms, why the evolution of $n(t)$ is described by

$\frac{d n}{d t}=-3 \frac{\dot{a}}{a} n-\langle\sigma v\rangle n \bar{n}+P(t)$

where the expansion scale factor of the universe is $a(t)$, and where the meaning of $P(t)$ should be briefly explained. Show that

$(n-\bar{n}) a^{3}=\text { constant }$

Assuming initial particle-antiparticle symmetry, show that

$\frac{d\left(n a^{3}\right)}{d t}=\langle\sigma v\rangle\left(n_{\mathrm{eq}}^{2}-n^{2}\right) a^{3}$

where $n_{\mathrm{eq}}$ is the equilibrium number density at temperature $T$.

Let $Y=n / T^{3}$ and $x=m / T$. Show that

$\frac{d Y}{d x}=-\frac{\lambda}{x^{2}}\left(Y^{2}-Y_{\mathrm{eq}}^{2}\right)$

where $\lambda=m^{3}\langle\sigma v\rangle / H_{m}$ and $H_{m}$ is the Hubble expansion rate when $T=m$.

When $x>x_{f} \simeq 10$, the number density $n$ can be assumed to be depleted only by annihilations. If $\lambda$ is constant, show that as $x \rightarrow \infty$ at late time, $Y$ approaches a constant value given by

$Y=\frac{x_{f}}{\lambda}$

Why do you expect weakly interacting particles to survive in greater numbers than strongly interacting particles?

Paper 4, Section I, C

commentCalculate the total effective number of relativistic spin states $g_{*}$ present in the early universe when the temperature $T$ is $10^{10} \mathrm{~K}$ if there are three species of low-mass neutrinos and antineutrinos in addition to photons, electrons and positrons. If the weak interaction rate is $\Gamma=\left(T / 10^{10} \mathrm{~K}\right)^{5} \mathrm{~s}^{-1}$ and the expansion rate of the universe is $H=\sqrt{8 \pi G \rho / 3}$, where $\rho$ is the total density of the universe, calculate the temperature $T_{*}$ at which the neutrons and protons cease to interact via weak interactions, and show that $T_{*} \propto g_{*}^{1 / 6}$.

State the formula for the equilibrium ratio of neutrons to protons at $T_{*}$, and briefly describe the sequence of events as the temperature falls from $T_{*}$ to the temperature at which the nucleosynthesis of helium and deuterium ends.

What is the effect of an increase or decrease of $g_{*}$ on the abundance of helium-4 resulting from nucleosynthesis? Why do changes in $g_{*}$ have a very small effect on the final abundance of deuterium?

Paper 1, Section II, $22 G$

commentLet $\Omega \subset \mathbb{R}^{2}$ be a domain (connected open subset) with boundary $\partial \Omega$ a continuously differentiable simple closed curve. Denoting by $A(\Omega)$ the area of $\Omega$ and $l(\partial \Omega)$ the length of the curve $\partial \Omega$, state and prove the isoperimetric inequality relating $A(\Omega)$ and $l(\partial \Omega)$ with optimal constant, including the characterization for equality. [You may appeal to Wirtinger's inequality as long as you state it precisely.]

Does the result continue to hold if the boundary $\partial \Omega$ is allowed finitely many points at which it is not differentiable? Briefly justify your answer by giving either a counterexample or an indication of a proof.

Paper 2, Section II, G

commentIf $U$ denotes a domain in $\mathbb{R}^{2}$, what is meant by saying that a smooth map $\phi: U \rightarrow \mathbb{R}^{3}$ is an immersion? Define what it means for such an immersion to be isothermal. Explain what it means to say that an immersed surface is minimal.

Let $\phi(u, v)=(x(u, v), y(u, v), z(u, v))$ be an isothermal immersion. Show that it is minimal if and only if $x, y, z$ are harmonic functions of $u, v$. [You may use the formula for the mean curvature given in terms of the first and second fundamental forms, namely $\left.H=(e G-2 f F+g E) /\left(2\left\{E G-F^{2}\right\}\right) .\right]$

Produce an example of an immersed minimal surface which is not an open subset of a catenoid, helicoid, or a plane. Prove that your example does give an immersed minimal surface in $\mathbb{R}^{3}$.

Paper 3 , Section II, G

commentShow that the surface $S$ of revolution $x^{2}+y^{2}=\cosh ^{2} z$ in $\mathbb{R}^{3}$ is homeomorphic to a cylinder and has everywhere negative Gaussian curvature. Show moreover the existence of a closed geodesic on $S$.

Let $S \subset \mathbb{R}^{3}$ be an arbitrary embedded surface which is homeomorphic to a cylinder and has everywhere negative Gaussian curvature. By using a suitable version of the Gauss-Bonnet theorem, show that $S$ contains at most one closed geodesic. [If required, appropriate forms of the Jordan curve theorem in the plane may also be used without proof.

Paper 4, Section II, G

commentLet $\mathrm{U}(n)$ denote the set of $n \times n$ unitary complex matrices. Show that $\mathrm{U}(n)$ is a smooth (real) manifold, and find its dimension. [You may use any general results from the course provided they are stated correctly.] For $A$ any matrix in $\mathrm{U}(n)$ and $H$ an $n \times n$ complex matrix, determine when $H$ represents a tangent vector to $\mathrm{U}(n)$ at $A$.

Consider the tangent spaces to $\mathrm{U}(n)$ equipped with the metric induced from the standard (Euclidean) inner product $\langle\cdot, \cdot\rangle$ on the real vector space of $n \times n$ complex matrices, given by $\langle L, K\rangle=\operatorname{Re} \operatorname{trace}\left(L K^{*}\right)$, where $\operatorname{Re}$ denotes the real part and $K^{*}$ denotes the conjugate transpose of $K$. Suppose that $H$ represents a tangent vector to $\mathrm{U}(n)$ at the identity matrix $I$. Sketch an explicit construction of a geodesic curve on $\mathrm{U}(n)$ passing through $I$ and with tangent direction $H$, giving a brief proof that the acceleration of the curve is always orthogonal to the tangent space to $\mathrm{U}(n)$.

[Hint: You will find it easier to work directly with $n \times n$ complex matrices, rather than the corresponding $2 n \times 2 n$ real matrices.]

Paper 1, Section II, 28B

comment(a) What is a Lyapunov function?

Consider the dynamical system for $\mathbf{x}(t)=(x(t), y(t))$ given by

$\begin{aligned} &\dot{x}=-x+y+x\left(x^{2}+y^{2}\right) \\ &\dot{y}=-y-2 x+y\left(x^{2}+y^{2}\right) \end{aligned}$

Prove that the origin is asymptotically stable (quoting carefully any standard results that you use).

Show that the domain of attraction of the origin includes the region $x^{2}+y^{2}<r_{1}^{2}$ where the maximum possible value of $r_{1}$ is to be determined.

Show also that there is a region $E=\left\{\mathbf{x} \mid x^{2}+y^{2}>r_{2}^{2}\right\}$ such that $\mathbf{x}(0) \in E$ implies that $|\mathbf{x}(t)|$ increases without bound. Explain your reasoning carefully. Find the smallest possible value of $r_{2}$.

(b) Now consider the dynamical system

$\begin{aligned} \dot{x} &=x-y-x\left(x^{2}+y^{2}\right) \\ \dot{y} &=y+2 x-y\left(x^{2}+y^{2}\right) \end{aligned}$

Prove that this system has a periodic solution (again, quoting carefully any standard results that you use).

Demonstrate that this periodic solution is unique.

Paper 2, Section II, B

comment(a) An autonomous dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{2}$ has a periodic orbit $\mathbf{x}=\mathbf{X}(t)$ with period $T$. The linearized evolution of a small perturbation $\mathbf{x}=\mathbf{X}(t)+\boldsymbol{\eta}(t)$ is given by $\eta_{i}(t)=\Phi_{i j}(t) \eta_{j}(0)$. Obtain the differential equation and initial condition satisfied by the matrix $\Phi(t)$.

Define the Floquet multipliers of the orbit. Explain why one of the multipliers is always unity and give a brief argument to show that the other is given by

$\exp \left(\int_{0}^{T} \nabla \cdot \mathbf{f}(\mathbf{X}(t)) d t\right)$

(b) Use the energy-balance method for nearly Hamiltonian systems to find leading-order approximations to the two limit cycles of the equation

$\ddot{x}+\epsilon\left(2 \dot{x}^{3}+2 x^{3}-4 x^{4} \dot{x}-\dot{x}\right)+x=0$

where $0<\epsilon \ll 1$.

Determine the stability of each limit cycle, giving reasoning where necessary.

[You may assume that $\int_{0}^{2 \pi} \cos ^{4} \theta d \theta=3 \pi / 4$ and $\int_{0}^{2 \pi} \cos ^{6} \theta d \theta=5 \pi / 8$.]

Paper 3, Section II, B

commentConsider the dynamical system

$\begin{aligned} &\dot{x}=-\mu+x^{2}-y \\ &\dot{y}=y(a-x) \end{aligned}$

where $a$ is to be regarded as a fixed real constant and $\mu$ as a real parameter.

Find the fixed points of the system and determine the stability of the system linearized about the fixed points. Hence identify the values of $\mu$ at given $a$ where bifurcations occur.

Describe informally the concepts of centre manifold theory and apply it to analyse the bifurcations that occur in the above system with $a=1$. In particular, for each bifurcation derive an equation for the dynamics on the extended centre manifold and hence classify the bifurcation.

What can you say, without further detailed calculation, about the case $a=0$ ?

Paper 4, Section II, B

commentLet $f: I \rightarrow I$ be a continuous one-dimensional map of an interval $I \subset \mathbb{R}$. Explain what is meant by the statements (i) that $f$ has a horseshoe and (ii) that $f$ is chaotic (according to Glendinning's definition).

Assume that $f$ has a 3-cycle $\left\{x_{0}, x_{1}, x_{2}\right\}$ with $x_{1}=f\left(x_{0}\right), x_{2}=f\left(x_{1}\right), x_{0}=f\left(x_{2}\right)$ and, without loss of generality, $x_{0}<x_{1}<x_{2}$. Prove that $f^{2}$ has a horseshoe. [You may assume the intermediate value theorem.]

Represent the effect of $f$ on the intervals $I_{a}=\left[x_{0}, x_{1}\right]$ and $I_{b}=\left[x_{1}, x_{2}\right]$ by means of a directed graph, explaining carefully how the graph is constructed. Explain what feature of the graph implies the existence of a 3-cycle.

The map $g: I \rightarrow I$ has a 5-cycle $\left\{x_{0}, x_{1}, x_{2}, x_{3}, x_{4}\right\}$ with $x_{i+1}=g\left(x_{i}\right), 0 \leqslant i \leqslant 3$ and $x_{0}=g\left(x_{4}\right)$, and $x_{0}<x_{1}<x_{2}<x_{3}<x_{4}$. For which $n, 1 \leqslant n \leqslant 4$, is an $n$-cycle of $g$ guaranteed to exist? Is $g$ guaranteed to be chaotic? Is $g$ guaranteed to have a horseshoe? Justify your answers. [You may use a suitable directed graph as part of your arguments.]

How do your answers to the above change if instead $x_{4}<x_{2}<x_{1}<x_{3}<x_{0}$ ?

Paper 1, Section II, A

commentBriefly explain how to interpret the components of the relativistic stress-energy tensor in terms of the density and flux of energy and momentum in some inertial frame.

(i) The stress-energy tensor of the electromagnetic field is

$T_{\mathrm{em}}^{\mu \nu}=\frac{1}{\mu_{0}}\left(F^{\mu \alpha} F_{\alpha}^{\nu}-\frac{1}{4} \eta^{\mu \nu} F^{\alpha \beta} F_{\alpha \beta}\right)$

where $F_{\mu \nu}$ is the field strength, $\eta_{\mu \nu}$ is the Minkowski metric, and $\mu_{0}$ is the permeability of free space. Show that $\partial_{\mu} T_{\mathrm{em}}^{\mu \nu}=-F_{\mu}^{\nu} J^{\mu}$, where $J^{\mu}$ is the current 4-vector.

[ Maxwell's equations are $\partial_{\mu} F^{\mu \nu}=-\mu_{0} J^{\nu}$ and $\partial_{\rho} F_{\mu \nu}+\partial_{\nu} F_{\rho \mu}+\partial_{\mu} F_{\nu \rho}=0 .$ ]

(ii) A fluid consists of point particles of rest mass $m$ and charge $q$. The fluid can be regarded as a continuum, with 4 -velocity $u^{\mu}(x)$ depending on the position $x$ in spacetime. For each $x$ there is an inertial frame $S_{x}$ in which the fluid particles at $x$ are at rest. By considering components in $S_{x}$, show that the fluid has a current 4-vector field

$J^{\mu}=q n_{0} u^{\mu}$

and a stress-energy tensor

$T_{\text {fluid }}^{\mu \nu}=m n_{0} u^{\mu} u^{\nu},$

where $n_{0}(x)$ is the proper number density of particles (the number of particles per unit spatial volume in $S_{x}$ in a small region around $x$ ). Write down the Lorentz 4-force on a fluid particle at $x$. By considering the resulting 4 -acceleration of the fluid, show that the total stress-energy tensor is conserved, i.e.

$\partial_{\mu}\left(T_{\mathrm{em}}^{\mu \nu}+T_{\text {fluid }}^{\mu \nu}\right)=0 .$

Paper 3, Section II, 34A

(i) Consider the action

$S=-\frac{1}{4 \mu_{0} c} \int\left(F_{\mu \nu} F^{\mu \nu}+2 \lambda^{2} A_{\mu} A^{\mu}\right) d^{4} x+\frac{1}{c} \int A_{\mu} J^{\mu} d^{4} x$

where $A_{\mu}(x)$ is a 4-vector potential, $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$ is the field strength tensor, $J^{\mu}(x)$ is a conserved current, and $\lambda \geqslant 0$ is a constant. Derive the field equation

$\partial_{\mu} F^{\mu \nu}-\lambda^{2} A^{\nu}=-\mu_{0} J^{\nu} .$

For $\lambda=0$ the action $S$ describes standard electromagnetism. Show that in this case the theory is invariant under gauge transformations of the field $A_{\mu}(x)$, which you should define. Is the theory invariant under these same gauge transformations when $\lambda>0$ ?

Show that when $\lambda>0$ the field equation above implies

$\partial_{\mu} \partial^{\mu} A^{\nu}-\lambda^{2} A^{\nu}=-\mu_{0} J^{\nu}$

Under what circumstances does $(*)$ hold in the case $\lambda=0$ ?

(ii) Now suppose that $A_{\mu}(x)$ and $J_{\mu}(x)$ obeying $(*)$ reduce to static 3 -vectors $\mathbf{A}(\mathbf{x})$ and $\mathbf{J}(\mathbf{x})$ in some inertial frame. Show that there is a solution

$\mathbf{A}(\mathbf{x})=-\mu_{0} \int G\left(\left|\mathbf{x}-\mathbf{x}^{\prime}\right|\right) \mathbf{J}\left(\mathbf{x}^{\prime}\right) d^{3} \mathbf{x}^{\prime}$

for a suitable Green's function $G(R)$ with $G(R) \rightarrow$