Variational Principles

# Variational Principles

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Paper 1, Section I, D

commentLet $D$ be a bounded region of $\mathbb{R}^{2}$, with boundary $\partial D$. Let $u(x, y)$ be a smooth function defined on $D$, subject to the boundary condition that $u=0$ on $\partial D$ and the normalization condition that

$\int_{D} u^{2} d x d y=1$

Let $I[u]$ be the functional

$I[u]=\int_{D}|\nabla u|^{2} d x d y$

Show that $I[u]$ has a stationary value, subject to the stated boundary and normalization conditions, when $u$ satisfies a partial differential equation of the form

$\nabla^{2} u+\lambda u=0$

in $D$, where $\lambda$ is a constant.

Determine how $\lambda$ is related to the stationary value of the functional $I[u]$. $[$ Hint: Consider $\boldsymbol{\nabla} \cdot(u \boldsymbol{\nabla} u)$.]

Paper 2, Section II, D

commentA particle of unit mass moves in a smooth one-dimensional potential $V(x)$. Its path $x(t)$ is such that the action integral

$S[x]=\int_{a}^{b} L(x, \dot{x}) d t$

has a stationary value, where $a$ and $b>a$ are constants, a dot denotes differentiation with respect to time $t$

$L(x, \dot{x})=\frac{1}{2} \dot{x}^{2}-V(x)$

is the Lagrangian function and the initial and final positions $x(a)$ and $x(b)$ are fixed.

By considering $S[x+\epsilon \xi]$ for suitably restricted functions $\xi(t)$, derive the differential equation governing the motion of the particle and obtain an integral expression for the second variation $\delta^{2} S$.

If $x(t)$ is a solution of the equation of motion and $x(t)+\epsilon u(t)+O\left(\epsilon^{2}\right)$ is also a solution of the equation of motion in the limit $\epsilon \rightarrow 0$, show that $u(t)$ satisfies the equation

$\ddot{u}+V^{\prime \prime}(x) u=0$

If $u(t)$ satisfies this equation and is non-vanishing for $a \leqslant t \leqslant b$, show that

$\delta^{2} S=\frac{1}{2} \int_{a}^{b}\left(\dot{\xi}-\frac{\dot{u} \xi}{u}\right)^{2} d t$

Consider the simple harmonic oscillator, for which

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

where $2 \pi / \omega$ is the oscillation period. Show that the solution of the equation of motion is a local minimum of the action integral, provided that the time difference $b-a$ is less than half an oscillation period.

Paper 3, Section I, D

commentFind the function $y(x)$ that gives a stationary value of the functional

$I[y]=\int_{0}^{1}\left(y^{\prime 2}+y y^{\prime}+y^{\prime}+y^{2}+y x^{2}\right) d x$

subject to the boundary conditions $y(0)=-1$ and $y(1)=e-e^{-1}-\frac{3}{2}$.

Paper 4 , Section II, 13D

comment(a) Consider the functional

$I[y]=\int_{a}^{b} L\left(y, y^{\prime} ; x\right) d x$

where $0<a<b$, and $y(x)$ is subject to the requirement that $y(a)$ and $y(b)$ are some fixed constants. Derive the equation satisfied by $y(x)$ when $\delta I=0$ for all variations $\delta y$ that respect the boundary conditions.

(b) Consider the function

$L\left(y, y^{\prime} ; x\right)=\frac{\sqrt{1+y^{\prime 2}}}{x} .$

Verify that, if $y(x)$ describes an arc of a circle, with centre on the $y$-axis, then $\delta I=0$.

(c) Consider the function

$L\left(y, y^{\prime} ; x\right)=\frac{\sqrt{1+y^{\prime 2}}}{y}$

Find $y(x)$ such that $\delta I=0$ subject to the requirement that $y(a)=a$ and $y(b)=\sqrt{2 a b-b^{2}}$, with $b<2 a$. Sketch the curve $y(x)$.

Paper 1, Section II, D

commentA motion sensor sits at the origin, in the middle of a field. The probability that you are detected as you sneak from one point to another along a path $\mathbf{x}(t): 0 \leqslant t \leqslant T$ is

$P[\mathbf{x}(t)]=\lambda \int_{0}^{T} \frac{v(t)}{r(t)} d t$

where $\lambda$ is a positive constant, $r(t)$ is your distance to the sensor, and $v(t)$ is your speed. (If $P[\mathbf{x}(t)] \geqslant 1$ for some path then you are detected with certainty.)

You start at point $(x, y)=(A, 0)$, where $A>0$. Your mission is to reach the point $(x, y)=(B \cos \alpha, B \sin \alpha)$, where $B>0$. What path should you take to minimise the chance of detection? Should you tiptoe or should you run?

A new and improved sensor detects you with probability

$\tilde{P}[\mathbf{x}(t)]=\lambda \int_{0}^{T} \frac{v(t)^{2}}{r(t)} d t$

Show that the optimal path now satisfies the equation

$\left(\frac{d r}{d t}\right)^{2}=E r-h^{2}$

for some constants $E$ and $h$ that you should identify.

Paper 2, Section I, D

commentFind the stationary points of the function $\phi=x y z$ subject to the constraint $x+a^{2} y^{2}+z^{2}=b^{2}$, with $a, b>0$. What are the maximum and minimum values attained by $\phi$, subject to this constraint, if we further restrict to $x \geqslant 0$ ?

Paper 1, Section I, A

commentA function $\phi=x y-y z$ is defined on the surface $x^{2}+2 y^{2}+z^{2}=1$. Find the location $(x, y, z)$ of every stationary point of this function.

Paper 2, Section II, A

commentWrite down the Euler-Lagrange (EL) equations for a functional

$\int_{a}^{b} f\left(u, w, u^{\prime}, w^{\prime}, x\right) d x$

where $u(x)$ and $w(x)$ each take specified values at $x=a$ and $x=b$. Show that the EL equations imply that

$\kappa=f-u^{\prime} \frac{\partial f}{\partial u^{\prime}}-w^{\prime} \frac{\partial f}{\partial w^{\prime}}$

is independent of $x$ provided $f$ satisfies a certain condition, to be specified. State conditions under which there exist additional first integrals of the $\mathrm{EL}$ equations.

Consider

$f=\left(1-\frac{m}{u}\right) w^{\prime 2}-\left(1-\frac{m}{u}\right)^{-1} u^{\prime 2}$

where $m$ is a positive constant. Show that solutions of the EL equations satisfy

$u^{\prime 2}=\lambda^{2}+\kappa\left(1-\frac{m}{u}\right)$

for some constant $\lambda$. Assuming that $\kappa=-\lambda^{2}$, find $d w / d u$ and hence determine the most general solution for $w$ as a function of $u$ subject to the conditions $u>m$ and $w \rightarrow-\infty$ as $u \rightarrow \infty$. Show that, for any such solution, $w \rightarrow \infty$ as $u \rightarrow m$.

[Hint:

$\left.\frac{d}{d z}\left\{\log \left(\frac{z^{1 / 2}-1}{z^{1 / 2}+1}\right)\right\}=\frac{1}{z^{1 / 2}(z-1)} . \quad\right]$

Paper 3, Section I, A

commentThe function $f$ with domain $x>0$ is defined by $f(x)=\frac{1}{a} x^{a}$, where $a>1$. Verify that $f$ is convex, using an appropriate sufficient condition.

Determine the Legendre transform $f^{*}$ of $f$, specifying clearly its domain of definition, and find $\left(f^{*}\right)^{*}$.

Show that

$\frac{x^{r}}{r}+\frac{y^{s}}{s} \geqslant x y$

where $x, y>0$ and $r$ and $s$ are positive real numbers such that $\frac{1}{r}+\frac{1}{s}=1$.

Paper 4, Section II, A

commentConsider the functional

$I[y]=\int_{-\infty}^{\infty}\left(\frac{1}{2} y^{\prime 2}+\frac{1}{2} U(y)^{2}\right) d x$

where $y(x)$ is subject to boundary conditions $y(x) \rightarrow a_{\pm}$as $x \rightarrow \pm \infty$ with $U\left(a_{\pm}\right)=0$. [You may assume the integral converges.]

(a) Find expressions for the first-order and second-order variations $\delta I$ and $\delta^{2} I$ resulting from a variation $\delta y$ that respects the boundary conditions.

(b) If $a_{\pm}=a$, show that $I[y]=0$ if and only if $y(x)=a$ for all $x$. Explain briefly how this is consistent with your results for $\delta I$ and $\delta^{2} I$ in part (a).

(c) Now suppose that $U(y)=c^{2}-y^{2}$ with $a_{\pm}=\pm c(c>0)$. By considering an integral of $U(y) y^{\prime}$, show that

$I[y] \geqslant \frac{4 c^{3}}{3},$

with equality if and only if $y$ satisfies a first-order differential equation. Deduce that global minima of $I[y]$ with the specified boundary conditions occur precisely for

$y(x)=c \tanh \left\{c\left(x-x_{0}\right)\right\}$

where $x_{0}$ is a constant. How is the first-order differential equation that appears in this case related to your general result for $\delta I$ in part (a)?

Paper 1, Section I, B

commentFind, using a Lagrange multiplier, the four stationary points in $\mathbb{R}^{3}$ of the function $x^{2}+y^{2}+z^{2}$ subject to the constraint $x^{2}+2 y^{2}-z^{2}=1$. By sketching sections of the constraint surface in each of the coordinate planes, or otherwise, identify the nature of the constrained stationary points.

How would the location of the stationary points differ if, instead, the function $x^{2}+2 y^{2}-z^{2}$ were subject to the constraint $x^{2}+y^{2}+z^{2}=1 ?$

Paper 2, Section II, B

commentDerive the Euler-Lagrange equation for the integral

$I[y]=\int_{x_{0}}^{x_{1}} f\left(y, y^{\prime}, y^{\prime \prime}, x\right) d x$

when $y(x)$ and $y^{\prime}(x)$ take given values at the fixed endpoints.

Show that the only function $y(x)$ with $y(0)=1, y^{\prime}(0)=2$ and $y(x) \rightarrow 0$ as $x \rightarrow \infty$ for which the integral

$I[y]=\int_{0}^{\infty}\left(y^{2}+\left(y^{\prime}\right)^{2}+\left(y^{\prime}+y^{\prime \prime}\right)^{2}\right) d x$

is stationary is $(3 x+1) e^{-x}$.

Paper 3, Section I, B

commentFor a particle of unit mass moving freely on a unit sphere, the Lagrangian in polar coordinates is

$L=\frac{1}{2} \dot{\theta}^{2}+\frac{1}{2} \sin ^{2} \theta \dot{\phi}^{2}$

Determine the equations of motion. Show that $l=\sin ^{2} \theta \dot{\phi}$ is a conserved quantity, and use this result to simplify the equation of motion for $\theta$. Deduce that

$h=\dot{\theta}^{2}+\frac{l^{2}}{\sin ^{2} \theta}$

is a conserved quantity. What is the interpretation of $h$ ?

Paper 4, Section II, B

comment(a) A two-dimensional oscillator has action

$S=\int_{t_{0}}^{t_{1}}\left\{\frac{1}{2} \dot{x}^{2}+\frac{1}{2} \dot{y}^{2}-\frac{1}{2} \omega^{2} x^{2}-\frac{1}{2} \omega^{2} y^{2}\right\} d t$

Find the equations of motion as the Euler-Lagrange equations associated with $S$, and use them to show that

$J=\dot{x} y-\dot{y} x$

is conserved. Write down the general solution of the equations of motion in terms of $\sin \omega t$ and $\cos \omega t$, and evaluate $J$ in terms of the coefficients that arise in the general solution.

(b) Another kind of oscillator has action

$\widetilde{S}=\int_{t_{0}}^{t_{1}}\left\{\frac{1}{2} \dot{x}^{2}+\frac{1}{2} \dot{y}^{2}-\frac{1}{4} \alpha x^{4}-\frac{1}{2} \beta x^{2} y^{2}-\frac{1}{4} \alpha y^{4}\right\} d t$

where $\alpha$ and $\beta$ are real constants. Find the equations of motion and use these to show that in general $J=\dot{x} y-\dot{y} x$ is not conserved. Find the special value of the ratio $\beta / \alpha$ for which $J$ is conserved. Explain what is special about the action $\widetilde{S}$in this case, and state the interpretation of $J$.

Paper 1, Section I, D

commentDerive the Euler-Lagrange equation for the function $u(x, y)$ that gives a stationary value of

$I[u]=\int_{\mathcal{D}} L\left(x, y, u, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}\right) d x d y$

where $\mathcal{D}$ is a bounded domain in the $(x, y)$-plane and $u$ is fixed on the boundary $\partial \mathcal{D}$.

Find the equation satisfied by the function $u$ that gives a stationary value of

$I=\int_{\mathcal{D}}\left[\left(\frac{\partial u}{\partial x}\right)^{2}+\left(\frac{\partial u}{\partial y}\right)^{2}+k^{2} u^{2}\right] d x d y$

where $k$ is a constant and $u$ is prescribed on $\partial \mathcal{D}$.

Paper 2, Section II, D

commentA proto-planet of mass $m$ in a uniform galactic dust cloud has kinetic and potential energies

$T=\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\phi}^{2}, \quad V=k m r^{2}$

where $k$ is constant. State Hamilton's principle and use it to determine the equations of motion for the proto-planet.

Write down two conserved quantities of the motion and state why their existence illustrates Noether's theorem.

Determine the Hamiltonian $H(\mathbf{p}, \mathbf{x})$ of this system, where $\mathbf{p}=\left(p_{r}, p_{\phi}\right), \mathbf{x}=(r, \phi)$ and $\left(p_{r}, p_{\phi}\right)$ are the conjugate momenta corresponding to $(r, \phi)$.

Write down Hamilton's equations for this system and use them to show that

$m \ddot{r}=-V_{\text {eff }}^{\prime}(r), \quad \text { where } \quad V_{\text {eff }}(r)=m\left(\frac{h^{2}}{2 m^{2} r^{2}}+k r^{2}\right)$

and $h$ is a constant. With the aid of a diagram, explain why there is a stable circular orbit.

Paper 3, Section $I$, D

comment(a) A Pringle crisp can be defined as the surface

$z=x y \quad \text { with } \quad x^{2}+y^{2} \leqslant 1$

Use the method of Lagrange multipliers to find the minimum and maximum values of $z$ on the boundary of the Pringle crisp and the $(x, y)$ positions where these occur.

(b) A farmer wishes to construct a grain silo in the form of a hollow vertical cylinder of radius $r$ and height $h$ with a hollow hemispherical cap of radius $r$ on top of the cylinder. The walls of the cylinder cost $£ x$ per unit area to construct and the surface of the cap costs $£ 2 x$ per unit area to construct. Given that a total volume $V$ is desired for the silo, what values of $r$ and $h$ should be chosen to minimise the cost?

Paper 4, Section II, $16 \mathrm{D}$

commentConsider the functional

$F[y]=\int_{\alpha}^{\beta} f\left(y, y^{\prime}, x\right) d x$

of a function $y(x)$ defined for $x \in[\alpha, \beta]$, with $y$ having fixed values at $x=\alpha$ and $x=\beta$.

By considering $F[y+\epsilon \xi]$, where $\xi(x)$ is an arbitrary function with $\xi(\alpha)=\xi(\beta)=0$ and $\epsilon \ll 1$, determine that the second variation of $F$ is

$\delta^{2} F[y, \xi]=\int_{\alpha}^{\beta}\left\{\xi^{2}\left[\frac{\partial^{2} f}{\partial y^{2}}-\frac{d}{d x}\left(\frac{\partial^{2} f}{\partial y \partial y^{\prime}}\right)\right]+\xi^{\prime 2} \frac{\partial^{2} f}{\partial y^{\prime 2}}\right\} d x$

The surface area of an axisymmetric soap film joining two parallel, co-axial, circular rings of radius a distance $2 L$ apart can be expressed by the functional

$F[y]=\int_{-L}^{L} 2 \pi y \sqrt{1+y^{\prime 2}} d x$

where $x$ is distance in the axial direction and $y$ is radial distance from the axis. Show that the surface area is stationary when

$y=E \cosh \frac{x}{E},$

where $E$ is a constant that need not be determined, and that the stationary area is a local minimum if

$\int_{-L / E}^{L / E}\left(\xi^{\prime 2}-\xi^{2}\right) \operatorname{sech}^{2} z d z>0$

for all functions $\xi(z)$ that vanish at $z=\pm L / E$, where $z=x / E$.

Paper 1, Section I, C

comment(a) Consider the function $f\left(x_{1}, x_{2}\right)=2 x_{1}^{2}+x_{2}^{2}+\alpha x_{1} x_{2}$, where $\alpha$ is a real constant. For what values of $\alpha$ is the function $f$ convex?

(b) In the case $\alpha=-3$, calculate the extremum of $x_{1}^{2}$ on the set of points where $f\left(x_{1}, x_{2}\right)+1=0 .$

Paper 2, Section II, C

commentA flexible wire filament is described by the curve $(x, y(x), z(x))$ in cartesian coordinates for $0 \leqslant x \leqslant L$. The filament is assumed to be almost straight and thus we assume $\left|y^{\prime}\right| \ll 1$ and $\left|z^{\prime}\right| \ll 1$ everywhere.

(a) Show that the total length of the filament is approximately $L+\Delta$ where

$\Delta=\frac{1}{2} \int_{0}^{L}\left[\left(y^{\prime}\right)^{2}+\left(z^{\prime}\right)^{2}\right] d x$

(b) Under a uniform external axial force, $F>0$, the filament adopts the shape which minimises the total energy, $\mathcal{E}=E_{B}-F \Delta$, where $E_{B}$ is the bending energy given by

$E_{B}[y, z]=\frac{1}{2} \int_{0}^{L}\left[A(x)\left(y^{\prime \prime}\right)^{2}+B(x)\left(z^{\prime \prime}\right)^{2}\right] d x$

and where $A(x)$ and $B(x)$ are $x$-dependent bending rigidities (both known and strictly positive). The filament satisfies the boundary conditions

$y(0)=y^{\prime}(0)=z(0)=z^{\prime}(0)=0, \quad y(L)=y^{\prime}(L)=z(L)=z^{\prime}(L)=0$

Derive the Euler-Lagrange equations for $y(x)$ and $z(x)$.

(c) In the case where $A=B=1$ and $L=1$, show that below a critical force, $F_{c}$, which should be determined, the only energy-minimising solution for the filament is straight $(y=z=0)$, but that a new nonzero solution is admissible at $F=F_{c}$.

Paper 3, Section I, C

commentTwo points $A$ and $B$ are located on the curved surface of the circular cylinder of radius $R$ with axis along the $z$-axis. We denote their locations by $\left(R, \phi_{A}, z_{A}\right)$ and $\left(R, \phi_{B}, z_{B}\right)$ using cylindrical polar coordinates and assume $\phi_{A} \neq \phi_{B}, z_{A} \neq z_{B}$. A path $\phi(z)$ is drawn on the cylinder to join $A$ and $B$. Show that the path of minimum distance between the points $A$ and $B$ is a helix, and determine its pitch. [For a helix with axis parallel to the $z$ axis, the pitch is the change in $z$ after one complete helical turn.]

Paper 4, Section II, C

commentA fish swims in the ocean along a straight line with speed $V(t)$. The fish starts its journey from rest (zero velocity at $t=0$ ) and, during a given time $T$, swims subject to the constraint that the total distance travelled is $L$. The energy cost for swimming is $a V^{2}+b \dot{V}^{2}$ per unit time, where $a, b \geqslant 0$ are known and $a^{2}+b^{2} \neq 0$.

(a) Derive the Euler-Lagrange condition on $V(t)$ for the journey to have minimum energetic cost.

(b) In the case $a \neq 0, b \neq 0$ solve for $V(t)$ assuming that the fish starts at $t=0$ with zero acceleration (in addition to zero velocity).

(c) In the case $a=0$, the fish can decide between three different boundary conditions for its journey. In addition to starting with zero velocity, it can:

(1) start at $t=0$ with zero acceleration;

(2) end at $t=T$ with zero velocity; or

(3) end at $t=T$ with zero acceleration.

Which of $(1),(2)$ or (3) is the best minimal-energy cost strategy?

Paper 1, Section I, A

commentConsider a frictionless bead on a stationary wire. The bead moves under the action of gravity acting in the negative $y$-direction and the wire traces out a path $y(x)$, connecting points $(x, y)=(0,0)$ and $(1,0)$. Using a first integral of the Euler-Lagrange equations, find the choice of $y(x)$ which gives the shortest travel time, starting from rest. You may give your solution for $y$ and $x$ separately, in parametric form.

Paper 2, Section II, A

commentA right circular cylinder of radius $a$ and length $l$ has volume $V$ and total surface area $A$. Use Lagrange multipliers to do the following:

(a) Show that, for a given total surface area, the maximum volume is

$V=\frac{1}{3} \sqrt{\frac{A^{3}}{C \pi}}$

determining the integer $C$ in the process.

(b) For a cylinder inscribed in the unit sphere, show that the value of $l / a$ which maximises the area of the cylinder is

$D+\sqrt{E}$

determining the integers $D$ and $E$ as you do so.

(c) Consider the rectangular parallelepiped of largest volume which fits inside a hemisphere of fixed radius. Find the ratio of the parallelepiped's volume to the volume of the hemisphere.

[You need not show that suitable extrema you find are actually maxima.]

Paper 3, Section $I$, A

comment(a) Define what it means for a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ to be convex.

(b) Define the Legendre transform $f^{*}(p)$ of a convex function $f(x)$, where $x \in \mathbb{R}$. Show that $f^{*}(p)$ is a convex function.

(c) Find the Legendre transform $f^{*}(p)$ of the function $f(x)=e^{x}$, and the domain of $f^{*}$.

Paper 4, Section II, A

commentDerive the Euler-Lagrange equation for the integral

$\int_{x_{0}}^{x_{1}} f\left(x, u, u^{\prime}\right) d x$

where $u\left(x_{0}\right)$ is allowed to float, $\partial f /\left.\partial u^{\prime}\right|_{x_{0}}=0$ and $u\left(x_{1}\right)$ takes a given value.

Given that $y(0)$ is finite, $y(1)=1$ and $y^{\prime}(1)=1$, find the stationary value of

$J=\int_{0}^{1}\left(x^{4}\left(y^{\prime \prime}\right)^{2}+4 x^{2}\left(y^{\prime}\right)^{2}\right) d x$

Paper 1, Section I, C

commentDefine the Legendre transform $f^{*}(\mathbf{p})$ of a function $f(\mathbf{x})$ where $\mathbf{x} \in \mathbb{R}^{n}$.

Show that for $g(\mathbf{x})=\lambda f\left(\mathbf{x}-\mathbf{x}_{0}\right)-\mu$,

$g^{*}(\mathbf{p})=\lambda f^{*}\left(\frac{\mathbf{p}}{\lambda}\right)+\mathbf{p}^{\mathbf{T}} \mathbf{x}_{0}+\mu$

Show that for $f(\mathbf{x})=\frac{1}{2} \mathbf{x}^{\mathbf{T}} \mathbf{A} \mathbf{x}$ where $\mathbf{A}$ is a real, symmetric, invertible matrix with positive eigenvalues,

$f^{*}(\mathbf{p})=\frac{1}{2} \mathbf{p}^{\mathbf{T}} \mathbf{A}^{-\mathbf{1}} \mathbf{p}$

Paper 2, Section II, C

commentWrite down the Euler-Lagrange equation for the integral

$\int f\left(y, y^{\prime}, x\right) d x$

An ant is walking on the surface of a sphere, which is parameterised by $\theta \in[0, \pi]($ angle from top of sphere) and $\phi \in[0,2 \pi$ ) (azimuthal angle). The sphere is sticky towards the top and the bottom and so the ant's speed is proportional to $\sin \theta$. Show that the ant's fastest route between two points will be of the form

$\sinh (A \phi+B)=\cot \theta$

for some constants $A$ and $B$. $[A, B$ need not be determined.]

Paper 3, Section I, $\mathbf{6 C}$

commentLet $f(x, y, z)=x z+y z$. Using Lagrange multipliers, find the location(s) and value of the maximum of $f$ on the intersection of the unit sphere $\left(x^{2}+y^{2}+z^{2}=1\right)$ and the ellipsoid given by $\frac{1}{4} x^{2}+\frac{1}{4} y^{2}+4 z^{2}=1$.

Paper 4, Section II, C

commentConsider the integral

$I=\int f\left(y, y^{\prime}\right) d x$

Show that if $f$ satisfies the Euler-Lagrange equation, then

$f-y^{\prime} \frac{\partial f}{\partial y^{\prime}}=\text { constant. }$

An axisymmetric soap film $y(x)$ is formed between two circular wires at $x=\pm l$. The wires both have radius $r$. Show that the shape that minimises the surface area takes the form

$y(x)=k \cosh \frac{x}{k}$

Show that there exist two possible $k$ that satisfy the boundary conditions for $r / l$ sufficiently large.

Show that for these solutions the second variation is given by

$\delta^{2} I=\pi \int_{-l}^{+l}\left(k \eta^{\prime 2}-\frac{1}{k} \eta^{2}\right) \operatorname{sech}^{2}\left(\frac{x}{k}\right) d x$

where $\eta$ is an axisymmetric perturbation with $\eta(\pm l)=0$.

Paper 1, Section I, A

comment(a) Define what it means for a function $g: \mathbb{R} \rightarrow \mathbb{R}$ to be convex. Assuming $g^{\prime \prime}$ exists, state an equivalent condition. Let $f(x)=x \log x$, defined on $x>0$. Show that $f(x)$ is convex.

(b) Find the Legendre transform $f^{*}(p)$ of $f(x)=x \log x$. State the domain of $f^{*}(p)$. Without further calculation, explain why $\left(f^{*}\right)^{*}=f$ in this case.

Paper 2, Section II, A

commentStarting from the Euler-Lagrange equation, show that a condition for

$\int f\left(y, y^{\prime}\right) d x$

to be stationary is

$f-y^{\prime} \frac{\partial f}{\partial y^{\prime}}=\text { constant }$

In the half-plane $y>0$, light has speed $c(y)=y+c_{0}$ where $c_{0}>0$. Find the equation for a light ray between $(-a, 0)$ and $(a, 0)$. Sketch the solution.

Paper 3, Section I, A

commentA cylindrical drinking cup has thin curved sides with density $\rho$ per unit area, and a disk-shaped base with density $k \rho$ per unit area. The cup has capacity to hold a fixed volume $V$ of liquid. Use the method of Lagrange multipliers to find the minimum mass of the cup.

Paper 4, Section II, A

commentDerive the Euler-Lagrange equation for the integral

$\int_{a}^{b} f\left(x, y, y^{\prime}, y^{\prime \prime}\right) d x$

where prime denotes differentiation with respect to $x$, and both $y$ and $y^{\prime}$ are specified at $x=a, b$.

Find $y(x)$ that extremises the integral

$\int_{0}^{\pi}\left(y+\frac{1}{2} y^{2}-\frac{1}{2} y^{\prime \prime 2}\right) d x$

subject to $y(0)=-1, y^{\prime}(0)=0, y(\pi)=\cosh \pi$ and $y^{\prime}(\pi)=\sinh \pi$.

Show that your solution is a global maximum. You may use the result that

$\int_{0}^{\pi} \phi^{2}(x) d x \leqslant \int_{0}^{\pi} \phi^{\prime 2}(x) d x$

for any (suitably differentiable) function $\phi$ which satisfies $\phi(0)=0$ and $\phi(\pi)=0$.

Paper 1, Section I, B

commentState how to find the stationary points of a $C^{1}$ function $f(x, y)$ restricted to the circle $x^{2}+y^{2}=1$, using the method of Lagrange multipliers. Explain why, in general, the method of Lagrange multipliers works, in the case where there is just one constraint.

Find the stationary points of $x^{4}+2 y^{3}$ restricted to the circle $x^{2}+y^{2}=1$.

Paper 2, Section II, B

comment(i) A two-dimensional oscillator has action

$S=\int_{t_{0}}^{t_{1}}\left\{\frac{1}{2} \dot{x}^{2}+\frac{1}{2} \dot{y}^{2}-\frac{1}{2} \omega^{2} x^{2}-\frac{1}{2} \omega^{2} y^{2}\right\} d t$

Find the equations of motion as the Euler-Lagrange equations associated to $S$, and use them to show that

$J=\dot{x} y-\dot{y} x$

is conserved. Write down the general solution of the equations of motion in terms of sin $\omega t$ and $\cos \omega t$, and evaluate $J$ in terms of the coefficients which arise in the general solution.

(ii) Another kind of oscillator has action

$\widetilde{S}=\int_{t_{0}}^{t_{1}}\left\{\frac{1}{2} \dot{x}^{2}+\frac{1}{2} \dot{y}^{2}-\frac{1}{4} \alpha x^{4}-\frac{1}{2} \beta x^{2} y^{2}-\frac{1}{4} \alpha y^{4}\right\} d t$

where $\alpha$ and $\beta$ are real constants. Find the equations of motion and use these to show that in general $J=\dot{x} y-\dot{y} x$ is not conserved. Find the special value of the ratio $\beta / \alpha$ for which $J$ is conserved. Explain what is special about the action $\widetilde{S}$in this case, and state the interpretation of $J$.

Paper 3, Section I, B

commentFor a particle of unit mass moving freely on a unit sphere, the Lagrangian in polar coordinates is

$L=\frac{1}{2} \dot{\theta}^{2}+\frac{1}{2} \sin ^{2} \theta \dot{\phi}^{2} .$

Find the equations of motion. Show that $l=\sin ^{2} \theta \dot{\phi}$ is a conserved quantity, and use this result to simplify the equation of motion for $\theta$. Deduce that

$h=\dot{\theta}^{2}+\frac{l^{2}}{\sin ^{2} \theta}$

is a conserved quantity. What is the interpretation of $h$ ?

Paper 4, Section II, B

commentConsider a functional

$I=\int_{a}^{b} F\left(x, y, y^{\prime}\right) d x$

where $F$ is smooth in all its arguments, $y(x)$ is a $C^{1}$ function and $y^{\prime}=\frac{d y}{d x}$. Consider the function $y(x)+h(x)$ where $h(x)$ is a small $C^{1}$ function which vanishes at $a$ and $b$. Obtain formulae for the first and second variations of $I$ about the function $y(x)$. Derive the Euler-Lagrange equation from the first variation, and state its variational interpretation.

Suppose now that

$I=\int_{0}^{1}\left(y^{\prime 2}-1\right)^{2} d x$

where $y(0)=0$ and $y(1)=\beta$. Find the Euler-Lagrange equation and the formula for the second variation of $I$. Show that the function $y(x)=\beta x$ makes $I$ stationary, and that it is a (local) minimizer if $\beta>\frac{1}{\sqrt{3}}$.

Show that when $\beta=0$, the function $y(x)=0$ is not a minimizer of $I$.

Paper 1, Section I, D

comment(i) Write down the Euler-Lagrange equations for the volume integral

$\int_{V}(\nabla u \cdot \nabla u+12 u) d V$

where $V$ is the unit ball $x^{2}+y^{2}+z^{2} \leqslant 1$, and verify that the function $u(x, y, z)=x^{2}+y^{2}+z^{2}$ gives a stationary value of the integral subject to the condition $u=1$ on the boundary.

(ii) Write down the Euler-Lagrange equations for the integral

$\int_{0}^{1}\left(\dot{x}^{2}+\dot{y}^{2}+4 x+4 y\right) d t$

where the dot denotes differentiation with respect to $t$, and verify that the functions $x(t)=t^{2}, y(t)=t^{2}$ give a stationary value of the integral subject to the boundary conditions $x(0)=y(0)=0$ and $x(1)=y(1)=1$.

Paper 2, Section II, D

comment(i) Let $I[y]=\int_{0}^{1}\left(\left(y^{\prime}\right)^{2}-y^{2}\right) d x$, where $y$ is twice differentiable and $y(0)=y(1)=0$. Write down the associated Euler-Lagrange equation and show that the only solution is $y(x)=0$.

(ii) Let $J[y]=\int_{0}^{1}\left(y^{\prime}+y \tan x\right)^{2} d x$, where $y$ is twice differentiable and $y(0)=y(1)=$ 0 . Show that $J[y]=0$ only if $y(x)=0$.

(iii) Show that $I[y]=J[y]$ and deduce that the extremal value of $I[y]$ is a global minimum.

(iv) Use the second variation of $I[y]$ to verify that the extremal value of $I[y]$ is a local minimum.

(v) How would your answers to part (i) differ in the case $I[y]=\int_{0}^{2 \pi}\left(\left(y^{\prime}\right)^{2}-y^{2}\right) d x$, where $y(0)=y(2 \pi)=0$ ? Show that the solution $y(x)=0$ is not a global minimizer in this case. (You may use without proof the result $I[x(2 \pi-x)]=-\frac{8}{15}\left(2 \pi^{2}-5\right)$.) Explain why the arguments of parts (iii) and (iv) cannot be used.

Paper 3, Section I, D

commentFind, using a Lagrange multiplier, the four stationary points in $\mathbb{R}^{3}$ of the function $x^{2}+y^{2}+z^{2}$ subject to the constraint $x^{2}+2 y^{2}-z^{2}=1$. By considering the situation geometrically, or otherwise, identify the nature of the constrained stationary points.

How would your answers differ if, instead, the stationary points of the function $x^{2}+2 y^{2}-z^{2}$ were calculated subject to the constraint $x^{2}+y^{2}+z^{2}=1 ?$

Paper 4 , Section II, D

commentDerive the Euler-Lagrange equation for the integral

$\int_{x_{0}}^{x_{1}} f\left(y, y^{\prime}, y^{\prime \prime}, x\right) d x$

where the endpoints are fixed, and $y(x)$ and $y^{\prime}(x)$ take given values at the endpoints.

Show that the only function $y(x)$ with $y(0)=1, y^{\prime}(0)=2$ and $y(x) \rightarrow 0$ as $x \rightarrow \infty$ for which the integral

$\int_{0}^{\infty}\left(y^{2}+\left(y^{\prime}\right)^{2}+\left(y^{\prime}+y^{\prime \prime}\right)^{2}\right) d x$

is stationary is $(3 x+1) e^{-x}$.

Paper 1, Section I, D

comment(a) Define what it means for a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ to be convex and strictly convex.

(b) State a necessary and sufficient first-order condition for strict convexity of $f \in C^{1}\left(\mathbb{R}^{n}\right)$, and give, with proof, an example of a function which is strictly convex but with second derivative which is not everywhere strictly positive.

Paper 2, Section II, D

commentDescribe briefly the method of Lagrange multipliers for finding the stationary points of a function $f(x, y)$ subject to a constraint $\phi(x, y)=0$.

A tent manufacturer wants to maximize the volume of a new design of tent, subject only to a constant weight (which is directly proportional to the amount of fabric used). The models considered have either equilateral-triangular or semi-circular vertical crosssection, with vertical planar ends in both cases and with floors of the same fabric. Which shape maximizes the volume for a given area $A$ of fabric?

[Hint: $(2 \pi)^{-1 / 2} 3^{-3 / 4}(2+\pi)<1 .$ ]

Paper 3, Section I, D

commentDerive the Euler-Lagrange equation for the function $u(x, y)$ which gives a stationary value of

$I=\int_{\mathcal{D}} L\left(x, y, u, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}\right) d x d y$

where $\mathcal{D}$ is a bounded domain in the $(x, y)$ plane, with $u$ fixed on the boundary $\partial \mathcal{D}$.

Find the equation satisfied by the function $u$ which gives a stationary value of

$I=\int_{\mathcal{D}}\left[\left(\frac{\partial u}{\partial x}\right)^{2}+\left(\frac{\partial u}{\partial y}\right)^{2}\right] d x d y$

with $u$ given on $\partial \mathcal{D}$.

Paper 4, Section II, D

commentA function $\theta(\phi)$ with given values of $\theta\left(\phi_{1}\right)$ and $\theta\left(\phi_{2}\right)$ makes the integral

$I=\int_{\phi_{1}}^{\phi_{2}} \mathcal{L}\left(\theta, \theta^{\prime}\right) d \phi$

stationary with respect to small variations of $\theta$ which vanish at $\phi_{1}$ and $\phi_{2}$. Show that $\theta(\phi)$ satisfies the first integral of the Euler-Lagrange equation,

$\mathcal{L}\left(\theta, \theta^{\prime}\right)-\theta^{\prime}\left(\partial \mathcal{L} / \partial \theta^{\prime}\right)=C,$

for some constant $C$. You may state the Euler-Lagrange equation without proof.

It is desired to tow an iceberg across open ocean from a point on the Antarctic coast (longitude $\phi_{1}$ ) to a place in Australia (longitude $\phi_{2}$ ), to provide fresh water for irrigation. The iceberg will melt at a rate proportional to the difference between its temperature (the constant $T_{0}$, measured in degrees Celsius and therefore negative) and the sea temperature $T(\theta)>T_{0}$, where $\theta$ is the colatitude (the usual spherical polar coordinate $\left.\theta\right)$. Assume that the iceberg is towed at a constant speed along a path $\theta(\phi)$, where $\phi$ is the longitude. Given that the infinitesimal arc length on the unit sphere is $\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)^{1 / 2}$, show that the total ice melted along the path from $\phi_{1}$ to $\phi_{2}$ is proportional to

$I=\int_{\phi_{1}}^{\phi_{2}}\left(T(\theta)-T_{0}\right)\left(\theta^{\prime 2}+\sin ^{2} \theta\right)^{1 / 2} d \phi$

Now suppose that in the relevant latitudes, the sea temperature may be approximated by $T(\theta)=T_{0}(1+3 \tan \theta)$. (Note that $(1+3 \tan \theta)$ is negative in the relevant latitudes.) Show that any smooth path $\theta(\phi)$ which minimizes the total ice melted must satisfy

$\theta^{\prime 2}=\sin ^{2} \theta\left(\frac{1}{4} k^{2} \tan ^{2} \theta \sin ^{2} \theta-1\right),$

and hence that

$\sin ^{2} \theta=\frac{2}{1-\left(1+k^{2}\right)^{1 / 2} \sin 2\left(\phi-\phi_{0}\right)},$

where $k$ and $\phi_{0}$ are constants.

[Hint:

$\int \frac{d x}{x\left(\alpha^{2} x^{4}+x^{2}-1\right)^{1 / 2}}=\frac{1}{2} \arcsin \left[\frac{x^{2}-2}{x^{2}\left(1+4 \alpha^{2}\right)^{1 / 2}}\right]$