Mathematical Methods for Physics and Engineering: A Comprehensive Guide

K. F. Riley, M. P. Hobson, S. J. Bence

Chapter 22 Calculus of variations - all with Video Answers

Educators

Chapter Questions

Problem 1

A surface of revolution, whose equation in cylindrical polar coordinates is $\rho=$ $\rho(z)$, is bounded by the circles $\rho=a, z=\pm c(a>c) .$ Show that the function that makes the surface integral $I=\int \rho^{-1 / 2} \bar{d} S$ stationary with respect to small variations is given by $\rho(z)=k+z^{2} /(4 k)$, where $k=\left[a \pm\left(a^{2}-c^{2}\right)^{1 / 2}\right] / 2$

Christian Otero

Numerade Educator

01:01

Problem 2

Show that the lowest value of the integral
$$
\int_{A}^{B} \frac{\left(1+y^{\prime 2}\right)^{1 / 2}}{y} d x
$$
where $A$ is $(-1,1)$ and $B$ is $(1,1)$, is $2 \ln (1+\sqrt{2})$. Assume that the Euler-Lagrange equation gives a minimising curve.

Raj Bala

Numerade Educator

06:49

Problem 3

The refractive index $n$ of a medium is a function only of the distance $r$ from a fixed point $O$. Prove that the equation of a light ray, assumed to lie in a plane through $O$, travelling in the medium satisfies (in plane polar coordinates)
$$
\frac{1}{r^{2}}\left(\frac{d r}{d \phi}\right)^{2}=\frac{r^{2}}{a^{2}} \frac{n^{2}(r)}{n^{2}(a)}-1
$$
where $a$ is the distance of the ray from $O$ at the point at which $d r / d \phi=0$.
If $n=\left[1+\left(\alpha^{2} / r^{2}\right)\right]^{1 / 2}$ and the ray starts and ends far from $O$, find its deviation (the angle through which the ray is turned) if its minimum distance from $O$ is $a$.

Ameer Said

Numerade Educator

03:24

Problem 4

The Lagrangian for a $\pi$-meson is given by
$$
L(\mathbf{x}, t)=\frac{1}{2}\left(\dot{\phi}^{2}-|\nabla \phi|^{2}-\mu^{2} \phi^{2}\right)
$$
where $\mu$ is the meson mass and $\phi(\mathbf{x}, t)$ is its wavefunction. Assuming Hamilton's principle find the wave equation satisfied by $\phi$.

Katie Mcalpine

Numerade Educator

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Problem 5

(a) For a system described in terms of coordinates $q_{1}$ and $t$, show that if $t$ does not appear explicitly in the expressions for $x, y$ and $z\left(x=x\left(q_{i}, t\right)\right.$, etc. $)$ then the kinetic energy $T$ is a homogeneous quadratic function of the $\dot{q}_{1}$ (it may also involve the $q_{i}$ ). Deduce that $\sum_{i} \dot{q}_{i}\left(\partial T / \partial \dot{q}_{t}\right)=2 T$.
(b) Assuming that the forces acting on the system are derivable from a potential. $V$, show, by expressing $d T / d t$ in terms of $q_{1}$ and $\dot{q}_{1}$, that $d(T+V) / d t=0$.

Susan Hallstrom

Numerade Educator

01:28

Problem 6

For a system specified by the coordinates $q$ and $t$, show that the equation of motion is unchanged if the Lagrangian $L(q, \dot{q}, t)$ is replaced by
$$
L_{1}=L+\frac{d \phi(q, t)}{d t}
$$
where $\phi$ is an arbitrary function. Deduce that the equation of motion of a particle that moves in one dimension subject to a force $-d V(x) / d x$ ( $x$ being measured from a point $O$ ) is unchanged if $O$ is forced to move with a constant velocity $v$ $(x$ still being measured from $O)$.

Penny Riley

Numerade Educator

02:46

Problem 7

In cylindrical polar coordinates, the curve $(\rho(\theta), \theta, \alpha \rho(\theta))$ lies on the surface of the cone $z=\alpha \rho$. Show that geodesics (curves of minimum length joining two points) on the cone satisfy
$$
\rho^{4}=c^{2}\left[\beta^{2} \rho^{\prime 2}+\rho^{2}\right]
$$
where $c$ is an arbitrary constant, but $\beta$ has to have a particular value. Determine the form of $\rho(\theta)$ and hence find the equation of the shortest path on the cone between the points $\left(R,-\theta_{0}, \alpha R\right)$ and $\left(R, \theta_{0}, \alpha R\right)$. (You will find it useful to determine the form of the derivative of $\cos ^{-1}\left(u^{-1}\right)$.)

James Kiss

Numerade Educator

02:02

Problem 8

22.7 In cylindrical polar coordinates, the curve $(\rho(\theta), \theta, \alpha \rho(\theta))$ lies on the surface of the cone $z=\alpha \rho$. Show that geodesics (curves of minimum length joining two points) on the cone satisfy
$$
\rho^{4}=c^{2}\left[\beta^{2} \rho^{\prime 2}+\rho^{2}\right]
$$
where $c$ is an arbitrary constant, but $\beta$ has to have a particular value. Determine the form of $\rho(\theta)$ and hence find the equation of the shortest path on the cone between the points $\left(R,-\theta_{0}, \alpha R\right)$ and $\left(R, \theta_{0}, \alpha R\right)$. (You will find it useful to determine the form of the derivative of $\cos ^{-1}\left(u^{-1}\right)$.)
$22.8$ Derive the differential equations for the polar coordinates $r, \theta$ of a particle of unit mass moving in a field of potential $V(r)$. Find the form of $V$ if the path of the particle is given by $r=a \sin \theta$

Nick Johnson

Numerade Educator

05:54

Problem 9

You are provided with a line of length $\pi a / 2$ and negligible mass and some lead shot of total mass $M .$ Use a variational method to determine how the lead shot must be distributed along the line if the loaded line is to hang in a circular arc of radius $a$ when its ends are attached to two points at the same height. (Measure the distance $s$ along the line from its centre.)

Subash Charan

Numerade Educator

01:12

Problem 10

Extend the result of subsection $22.2 .2$ to the case of several dependent variables $y_{i}(x)$, showing that, if $x$ does not appear explicitly in the integrand, then a first integral of the Euler-Lagrange equations is
$$
F-\sum_{i=1}^{n} y_{i}^{\prime} \frac{\partial F}{\partial y_{i}^{\prime}}=\text { constant }
$$

Raj Bala

Numerade Educator

00:25

Problem 11

A general result is that light travels through a variable medium by a path which minimises the travel time (this is an alternative formulation of Fermat's principle). With respect to a particular cylindrical polar coordinate system $(\rho, \phi, z)$ the speed of light $v(\rho, \phi)$ is independent of $z .$ If the path of the light is parameterised as $\rho=\rho(z), \phi=\phi(z)$, use the result of the previous exercise to show that
$$
v^{2}\left(\rho^{\prime 2}+\rho^{2} \phi^{\prime 2}+1\right)
$$
is constant along the path.
For the particular case when $v=v(\rho)=b\left(a^{2}+\rho^{2}\right)^{1 / 2}$, show that the two EulerLagrange equations have a common solution in which the light travels along a helical path given by $\phi=A z+B, \rho=C$, provided that $A$ has a particular value.

Mayukh Banik

Numerade Educator

01:09

Problem 12

Light travels in a vertical $x z$-plane through a slab of material which lies between the planes $z=z_{0}$ and $z=2 z_{0}$ and in which the speed of light $v(z)=c_{0} z / z_{0}$. Using the alternative formulation of Fermat's principle given in the previous question, show that the ray paths are arcs of circles.

Deduce that, if a ray enters the material at $\left(0, z_{0}\right)$ at an angle to the vertical, $\pi / 2-\theta$, of more than $30^{\circ}$, it does not reach the far side of the slab.

Ranjeet Singh

Numerade Educator

05:35

Problem 13

A dam of capacity $V$ (less than $\left.\pi b^{2} h / 2\right)$ is to be constructed on level ground next to a long straight wall which runs from $(-b, 0)$ to $(b, 0) .$ This is to be achieved by joining the ends of a new wall, of height $h$, to those of the existing wall. Show that, in order to minimise the length $L$ of new wall to be built, it should form part of a circle, and that $L$ is then given by
$$
\int_{-b}^{b} \frac{d x}{\left(1-\lambda^{2} x^{2}\right)^{1 / 2}}
$$
where $\lambda$ is found from
$$
\frac{V}{h b^{2}}=\frac{\sin ^{-1} \mu}{\mu^{2}}-\frac{\left(1-\mu^{2}\right)^{1 / 2}}{\mu}
$$
and $u=\lambda b$

Keshav Singh

Numerade Educator

01:26

Problem 14

The Schwarzchild metric for the static field of a non-rotating spherically symmetric black hole of mass $M$ is
$$
(d s)^{2}=c^{2}\left(1-\frac{2 G M}{c^{2} r}\right)(d t)^{2}-\frac{d r^{2}}{1-2 G M /\left(c^{2} r\right)}-r^{2}(d \theta)^{2}-r^{2} \sin ^{2} \theta(d \phi)^{2}
$$
Considering only motion confined to the plane $\theta=\pi / 2$, and assuming that the path of a small test particle is such as to make $\int d s$ stationary, find two first integrals of the equations of motion. From their Newtonian limits, in which $G M / r, \dot{r}^{2}$ and $r^{2} \dot{\phi}^{2}$ are all $<c^{2}$, identify the constants of integration.

Khoobchandra Agrawal

Numerade Educator

01:54

Problem 15

In the brachistochrone problem of subsection $22.3 .4$ show that if the upper endpoint can lie anywhere on the curve $h(x, y)=0$ then the curve of quickest descent $y(x)$ meets $h(x, y)=0$ at right angles.

Suzanne W.

Numerade Educator

01:20

Problem 16

Use result $(22.27)$ to evaluate
$$
J=\int_{-1}^{1}\left(1-x^{2}\right) P_{m}^{\prime}(x) P_{n}^{\prime}(x) d x
$$
where $P_{m}(x)$ is a Legendre polynomial of order $m$.

Manik Pulyani

Numerade Educator

01:04

Problem 17

Determine the minimum value that the integral
$$
J=\int_{0}^{1}\left[x^{4}\left(y^{\prime \prime}\right)^{2}+4 x^{2}\left(y^{\prime}\right)^{2}\right] d x
$$
can have, given that $y$ is not singular at $x=0$ and that $y(1)=y^{\prime}(1)=1$. Assume that the Euler-Lagrange equation does give the lower limit, and verify retrospectively that your solution makes the first term on the LHS of equation (22.15) vanish.

Raj Bala

Numerade Educator

00:57

Problem 18

Show that $y^{\prime \prime}-x y+\lambda x^{2} y=0$ has a solution for which $y(0)=y(1)=0$ and $\lambda \leq 147 / 4$

Dorcas Attuabea Addo

Numerade Educator

02:56

Problem 19

Find an appropriate but simple trial function and use it to estimate the lowest eigenvalue $\lambda_{0}$ of Stokes' equation
$$
\frac{d^{2} y}{d x^{2}}+\lambda x y=0, \quad y(0)=y(\pi)=0
$$
Explain why your estimate must be strictly greater than $\lambda_{0}$ -

M Hassan Anwar

Numerade Educator

02:32

Problem 20

Estimate the lowest eigenvalue $\lambda_{0}$ of the equation
$$
\frac{d^{2} y}{d x^{2}}-x^{2} y+\lambda y=0, \quad y(-1)=y(1)=0
$$
using a quadratic trial function.

M Hassan Anwar

Numerade Educator

06:57

Problem 21

A drumskin is stretched across a fixed circular rim of radius $a$. Small transverse vibrations of the skin have an amplitude $z(\rho, \phi, t)$ that satisfies
$$
\nabla^{2} z=\frac{1}{c^{2}} \frac{\partial^{2} z}{\partial t^{2}}
$$
in plane polar coordinates. For a normal mode independent of azimuth, $z=$ $Z(\rho) \cos \omega t$, find the differential equation satisfied by $Z(\rho)$. By using a trial function of the form $a^{v}-\rho^{r}$, obtain an estimate for the lowest normal mode frequency. (The exact answer is $\left.(5.78)^{1 / 2} c / a\right)$

Ozenc Gungor

Numerade Educator

02:43

Problem 22

(a) Recast the problem of finding the lowest eigenvalue $\lambda_{0}$ of the equation
$$
\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+2 x \frac{d y}{d x}+\lambda y=0, \quad y(\pm 1)=0
$$
in variational form, and derive an approximation $\lambda_{1}$ to $\lambda_{0}$ by using the trial function $y_{1}(x)=1-x^{2}$
(b) Show that an improved estimate $\lambda_{2}$ is obtained by using $y_{2}(x)=\cos (\pi x / 2)$
(c) Prove that the estimate $\lambda(\gamma)$ obtained by taking $y_{1}(x)+\gamma y_{2}(x)$ as the trial function is
$$
\lambda(\gamma)=\frac{64 / 15+16 \gamma / \pi+\left(\pi^{2} / 3+1 / 2\right) \gamma^{2}}{16 / 15+64 \gamma / \pi^{3}+\gamma^{2}}
$$
Investigate $\lambda(\gamma)$ numerically as $\gamma$ is varied, or, more simply, show that $\lambda(1)=3.183$, a significant improvement on both $\lambda_{1}$ and $\lambda_{2-}$

Victor Salazar

Numerade Educator

02:32

Problem 23

For the boundary conditions given below, obtain a functional $\Lambda(y)$ whose stationary values give the eigenvalues of the equation
$$
(1+x) \frac{d^{2} y}{d x^{2}}+(2+x) \frac{d y}{d x}+\lambda y=0, \quad y(0)=0, y^{\prime}(2)=0
$$

M Hassan Anwar

Numerade Educator

02:47

Problem 24

The upper and lower surfaces of a film of liquid with surface energy per unit area (surface tension) equal to $\gamma$ and with density $\rho$ have equations $z=p(x)$ and $z=q(x)$ respectively. The film has a given volume $V$ (per unit depth in the $y$ direction) and lies in the region $-L<x<L$, with $p(0)=q(0)=p(L)=q(L)=0 .$ The total energy (per unit depth) of the film consists of its surface energy and its gravitational energy, and is expressed by
$$
E=\frac{1}{2} \int_{-L}^{L}\left(p^{2}-q^{2}\right) d x+\gamma \int_{-L}^{L}\left[\left(1+p^{\prime 2}\right)^{1 / 2}+\left(1+q^{\prime 2}\right)^{1 / 2}\right] d x
$$
(a) Express $V$ in terms of $p$ and $q$.
(b) Show that, if the total energy is minimised, $p$ and $q$ must satisfy
$$
\frac{p^{\prime 2}}{\left(1+p^{\prime 2}\right)^{1 / 2}}-\frac{q^{\prime 2}}{\left(1+q^{\prime 2}\right)^{1 / 2}}=\text { constant }
$$
(c) As an approximate solution, consider the equations
$$
p=a(L-|x|), \quad q=b(L-|x|)
$$
where $a$ and $b$ are sufficiently small that $a^{3}$ and $b^{3}$ can be neglected compared to unity. Find the values of $a$ and $b$ that minimise $E$.

Dorcas Attuabea Addo

Numerade Educator

09:12

Problem 25

This is an alternative approach to the example in section $22.8 .$ Using the notation of that section, the expectation value of the energy of the state $\psi$ is given by $\int \psi^{*} H \psi d v$. Denote the eigenfunctions of $H$ by $\varphi_{1}$, so that $H \varphi_{i}=E_{i} \psi_{i}$, and, since $H$ is self-adjoint (Hermitian), $\int \psi_{j}^{*} \psi_{i} d v=\delta_{i j-}$
(a) By writing any function $\psi$ as $\sum c_{j} \psi_{j}$ and following an argument similar to that in section $22.7$, show that
$$
E=\frac{\int \psi^{*} H \psi d v}{\int \psi^{*} \varphi d v} \geq E_{0}
$$
the energy of the lowest state. (This is the Rayleigh-Ritz principle.)
(b) Using the same trial function as in section $22.8, \varphi=\exp \left(-x x^{2}\right)$,show that the same result is obtained.

Abhijit Das

Numerade Educator

08:34

Problem 26

This is an extension to section $22.8$ and the previous question. With the groundstate (i.e. the lowest-energy) wavefunction as $\exp \left(-\alpha x^{2}\right)$, take as a trial function the orthogonal wave function $x^{2 n+1} \exp \left(-\alpha x^{2}\right)$, using the integer $n$ as a variable parameter. Use either Sturm-Liouville theory or the Rayleigh-Ritz principle to show that the energy of the second lowest state of a quantum harmonic oscillator is $\leq 3 \hbar \omega / 2$

Suzanne W.

Numerade Educator

02:27

Problem 27

The Hamiltonian $H$ for the hydrogen atom is
$$
-\frac{\hbar^{2}}{2 m} \nabla^{2}-\frac{q}{4 \pi \epsilon_{0} r}
$$
For a spherically symmetric state, as may be assumed for the ground state, the only relevant part of $\nabla^{2}$ is that involving differentiation with respect to $r$.
(a) Define the integrals $J_{n}$ by
$$
J_{n}=\int_{0}^{\infty} r^{n} e^{-2 \beta r} d r
$$

Adriano Chikande

Numerade Educator

04:10

Problem 28

A particle of mass $m$ moves in a one-dimensional potential well of the form
$$
V(x)=-\mu \frac{\hbar^{2} \alpha^{2}}{m} \operatorname{sech}^{2} \alpha x
$$
where $\mu$ and $\alpha$ are positive constants. As in exercise $22.27$, the expectation value $\langle E\rangle$ of the energy of the system is $\int \varphi^{*} H \varphi d x$, where the self-adjoint operator $H$ is given by $-\left(\hbar^{2} / 2 m\right) d^{2} / d x^{2}+V(x) .$ Using trial wavefunctions of the form $y=A$ sech $\beta x$, show the following:
(a) for $\mu=1$ there is an exact eigenfunction of $H$, with a corresponding $\langle E\rangle$ of half of the maximum depth of the well;
(b) for $\mu=6$ the "binding energy' of the ground state is at least $10 \hbar^{2} \alpha^{2} /(3 m)$.
(You will find it useful to note that for $u, v \geq 0$, sech $u$ sech $v \geq$ sech $(u+v)$ )

Salamat Ali

Numerade Educator

04:02

Problem 29

The Sturm-Liouville equation can be extended to two independent variables, $x$ and $z$, with little modification. In equation $(22.22) y^{\prime 2}$ is replaced by $(\nabla y)^{2}$ and the integrals of the various functions of $y(x, z)$ become two-dimensional, i.e. the infinitesimal is $d x d z$.

The vibrations of a trampoline 4 units long and 1 unit wide satisfy the equation
$$
\nabla^{2} y+k^{2} y=0
$$
By taking the simplest possible permissible polynomial as a trial function, show that the lowest mode of vibration has $k^{2} \leq 10.63$ and, by direct solution, that the actual value is $10.49$

Matt Just

Numerade Educator