MATHEMATICAL METHODS III
Calculus of Variations
1. Show that the element of distance on a unit sphere, parametrised using the azimuthal
angle θ, is
1/2
2
′ 2
ds = 1 + sin θ [φ ]
dθ
[Hint: Start from the result (x, y, z) = (sin θ cos φ, sin θ sin φ, cos θ), compute (dx, dy, dz)
and hence ds2 = dx2 + dy 2 + dz 2 ).]
2. You will need to revise Lagrange multipliers from last year (Mathematical Methods II).
Revise the solution to the following problem: The equilibrium probability distribution
minimises the Helmholtz free energy
X
X
Ei pi + kT
pi ln pi
F =
i
i
over the probabilities {pi } subject to the constraint I =
P
i
pi = 1.
Extremise the quantity F − λI, where λ is a Lagrange multiplier, to find the equilibrium
probability distribution.
Once you understand the use of the Lagrange multiplier in this problem, the problems
with integral constraints will seem straightforward. You just need to think of the integral
as the limit of a sum and everything will work.
3. Sketch light rays in a horizontally stratified medium for the cases when the velocity
varies is given by
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(i) v(y) = ,
y
(ii) v(y) =
√
y,
√
and (iii) v(y) = 1/ y
In all three cases it is actually possible to find closed form solutions for the rays. To test
your ability at solving ODE’s, you might like to have a go at finding these solutions.
4. In the lectures it was claimed that the area of a soap film defined by u = u(x, y) is
increased with respect to its value when flat (u = constant) by the amount
Z
(u2x + u2y )
dxdy,
δS =
2
for the case where ux and uy are both small (ie when the surface is close to being flat).
Derive this formula as follows. When x → x + dx and y → y + dy the point on the
surface of the film at (x, y, u(x, y)) moves to
(x, y, u(x, y)) + dx (1, 0, ux (x, y))
and
(x, y, u(x, y)) + dy (0, 1, uy (x, y))
respectively. The area element for a flat film would be (dS)0 = (dx, 0, 0) ∧ (0, dy, 0). For
the film at (x, y, u(x, y)) it becomes
dS = dx dy (1, 0, ux ) ∧ (0, 1, uy ).
Here ∧ denotes the vector cross-product. Show that
|dS|2 = (dxdy)2 1 + u2x + u2y
and show that for small |ux | and |uy | the change in area, δS, is as given at the start of the
question.
5. A soap film is attached to two circular wires at r = a, z = ±b in cylindrical polar
coordinates (r, θ, z). Assuming that the surface is cylindrically symmetric, show that the
area of the surface is given by
Z b
1/2
2πr(z) 1 + r′ (z)2
dz.
S=
−b
Write down the E-L equation governing r(z). Show that the solution is
z a
b
.
= cosh
, where
r(z) = c cosh
c
c
c
Show graphically that there is no solution for c if b/a is larger than a certain critical ratio.
What happens to the surface as b/a is increased from below this ratio to above it?
6. In the lectures we used the trial wavefunction
x+a
b+a
a−x
= A
a−b
φ(x) = A
−a<x<b
b<x<a
to estimate the lowest eigenvalue of the operator −d2 /dx2 for functions φ(x) with
φ(x ≤ −a) = φ(x ≥ a) = 0.
Estimate the lowest eigenvalue for this boundary value problem using the trial function
φ(x) = A(a2 − x2 ).
Remember to normalise the function. Although there are no variational parameters, you
should find that this function gets very close to the exact eigenvalue.
2
7. Show that ψ0 = exp(− 12 x2 ) is an eigenfunction of the operator
H=−
d2
+ x2
2
dx
acting on functions ψ(x) for which ψ → 0 as |x| → ∞, and find the corresponding
eigenvalue λ0 .
Now consider the trial function
ψb =
A(a2 − x2 ) |x| < a
0
|x| ≥ a.
Normalize the function to find the constant A. Minimise the expectation value of the
b ψi)
b with respect to a and compare your estimate with the exact
eigenvalue (hψ|H|
eigenvalue.
8. The (action) functional for a particle of mass, m, and charge, q, in a magnetic field,
B(r) = ∇ ∧ A(r), and electrostatic potential, φ(r), is
2
Z
mv
S = dt
+ qv.A(r) − qφ(r) .
2
Here v = (ẋ, ẏ, ż) is the velocity of the particle.
Write down the Euler-Lagrange equations for this integral and show that they can be
written:
mv̇ = q(E + v ∧ B).
[When working with the E-L equations you should take care to note that
∂Ai
dAi
=
+ (v.∇)Ai ,
dt
∂t
ie that a total derivative with respect to time is required. ]
9. During your studies here and at school the importance of using SI quantities has been
emphasised. What we haven’t emphasised, of course, is that people do not always practise
what they preach and often choose their system to suit their problem. (This is actually a
very reasonable thing to do. SI units are based around human scale quantities like metres,
seconds, Amps and kilograms, while quantum theorists want to use natural scales set by
the fundamental constants. Roughly speaking, in these units the equations of motion are
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found from the familiar ones by setting c = µ0 = ǫ0 = 1.) Here are Maxwell’s equations
as you have met them:
∇ ∧ BSI = µ0 jSI + µ0 ǫ0
∇ ∧ ESI = −
∂ESI
∂tSI
∂BSI
∂tSI
∇.BSI = 0
ρSI
.
∇.ESI =
ǫ0
Here they are in a rationalised system:
∂E
∂t
∂B
∇∧E = −
∂t
∇.B = 0
∇.E = ρ.
∇∧B = j+
Check that the transformation
√
ρSI
BSI
√
B = √ , E = ESI ǫ0 , ρ = √ , j = µ0 jSI , t = ctSI
µ0
ǫ0
takes quantities from the SI to the rationalised system. (Remember that ǫ0 µ0 = 1/c2 .)
Check that the continuity equation ∇.j + ∂ρ/∂t = 0 has the same form in both systems.
NOT FOR THE FAINT-HEARTED
Physicists in elementary particle physics usually go one step further by setting h̄ = 1.
Everything is usually written as a power of energy. For example, lengths and time are
scaled:
l
t
tSI
l → l′ = ,
t → t′ =
= .
h̄c
h̄c
h̄
They then both have the dimensions of inverse energy.
The quantities in Maxwell’s equations also get scaled. For example,
B → B′ = (h̄c)3/2 B =
√
(h̄c)3/2 BSI
and E → E′ = (h̄c)3/2 E = ǫ0 (h̄c)3/2 ESI .
√
µ0
Find the dimension (power of energy) of the scaled fields, B′ and E′ . Find the scaled
version of the current, j, and its dimension. Hence find the scaled version of the
conductivity (j′ = σ ′ E′ ) and its dimension.
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10. We want to show that extremising the action functional
Z
S =
d4 xL (xµ , Aµ , ∂ν Aµ ) where
L
1
= − η µα η νβ (∂α Aβ − ∂β Aα ) (∂µ Aν − ∂ν Aµ ) − j ν Aν
4
leads to Aµ which satisfy Maxwell’s equations. As in the lectures, Aµ = ηµν Aν and we
are using the summation convention (repeated indices are summed over). The functional
depends on four ‘independent’ variables, the space-time coordinates, and the four
‘dependent’ variables, which are the components of the four-potential Aµ = (φ, −A).
Here φ is the electrostatic potential and A the magnetic vector potential. The components
of the four-current, j µ = (ρ, j), are taken to be quantities imposed from outside.
Check that you can see why
∂
∂µ Aν = δρµ δσν
∂ (∂ρ Aσ )
where δxy denotes the Kronecker delta, and why
∂
η µα η νβ ∂α Aβ = η µρ η νσ .
∂ (∂ρ Aσ )
Now show that the Euler-Lagrange equations for S
∂L
∂L
= ∂ρ
∂Aσ
∂ (∂ρ Aσ )
are the same as Maxwell’s equations:
∂ρ F ρσ = j σ where F ρσ = ∂ ρ Aσ − ∂ σ Aρ .
11. A simple theory describes the pion of mass, m, using the field, φ(xµ ), which
extremizes the action functional: (remember h̄ and c will have been set equal to 1—see
previous questions)
Z
Z
4
µ
2 2
S = d x ∂µ φ ∂ φ − m φ = d4 x η µν ∂µ φ ∂ν φ − m2 φ2 .
Write down the Euler-Lagrange equation for this functional and confirm that it is
consistent with Bohr’s correspondence principle.
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12. Consider the functional (µ = 0, 1, 2, 3)
µ
S [x (λ)] =
Z
dλ gαβ
dxα dxβ
.
dλ dλ
Here gαβ (xµ ) is the generalization of the quantity ηαβ , introduced in the lectures, to the
case where it depends on position in space-time, xµ . Like ηαβ , it is symmetric, gαβ = gβα ,
and has a contravariant version, g αβ , where g αβ gβδ = δαδ .
Show that the E-L equations for this functional can be written (the dot notation means
differentiate with respect to λ)
ẍα + Γαγβ ẋβ ẋγ = 0,
(1)
where
1
Γαγβ = g αδ (gδγ,β + gβδ,γ − gγβ,δ )
2
and the comma notation denotes partial differentiation:
F ,ρ = ∂ρ F .
[Aside: In the theory of General Relativity, motion is controlled by the structure of
space-time. Objects move along geodesics, which are the equivalent of straight lines in
curved space-time. The notion of a force does not appear. The great circle on the surface
of the sphere, which was treated in the lectures, is an example of a geodesic in a 2D
curved space. The equation (1) is the equation for a geodesic in 4D space-time, where g is
called the metric tensor. It replaces F = ma as the equation of motion for a particle.]
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MATHEMATICAL METHODS III
Complex Variables
Questions 1-3 are revision. You should check that you are OK with these BEFORE the
lectures on this topic begin.
1. (a) Show that 1i = −i; (b) Write (1 + i)4 in the form a + ib. Plot all of (1 + i), (1 + i)2 ,
(1 + i)3 and (1 + i)4 in the Argand diagram.
2. Check that: (a) (z1 z2 )∗ = z1∗ z2∗ ; (b) zz ∗ = |z|2 ; (c) Re z = (z + z ∗ )/2;
(d) |z + z ′ | ≤ |z| + |z ′ |
3. Starting from the definitions
sin z =
eiz − e−iz
,
2i
cos z =
eiz + e−iz
2
and
sinh z =
ez − e−z
,
2
show: (a) sin2 z + cos2 z = 1; (b) sin(z + w) = sin z cos w + cos z sin w;
(c) sin iz = i sinh z.
4. Differentiate and give the appropriate domain of analyticity for each of the following
(z 2 + z)3 , 1/z and 1/ cos z 2 .
5. Determine whether the following limits exist and find their value if they do:
ez − 1
sin |z|
a) lim
, b) lim
z→0
z→0
z
z
6. Find analytic functions whose real parts are
a) y,
b)
x
2
x + y2
and c) tan−1
7
y
x
7. If f (z) = z 2 ≡ u(x, y) + iv(x, y) find u(x, y) and v(x, y). Draw contours on the
complex plane showing lines of constant u and constant v.
Show that these intersect at right angles. (Remember that contours of any function,
φ(x, y), are perpendicular to ∇φ.) Show using the Cauchy Riemann equations that lines of
constant u and lines of constant v intersect at right angles for all analytic functions.
8. Show that the transformation z 7→ w with w = (z − 1)/(z − 2) takes the line z =
onto a circle in the w-plane and find its radius.
3
2
+ iy
[Analytic functions define conformal mappings. These preserve the sense of angles and
have local magnification. Can you see why?]
9. Evaluate the integrals
Z
z dz and
Γ
Z
z ∗ dz
Γ
for the two contours shown:
z0 = (x0 , y0 )
z0 = (x0 , y0 )
Γ1
Γ2
10. For which simple closed curves C is
I
C
dz
=0?
z2 + z + 1
H
11. Find C (z − a)m dz where C is any closed contour about z = a and m is a positive or
negative integer.
12.
a. Expand f (z) = 1/(1 − z) about (i) z = 0 and ii) z = −1. Draw the circle of
convergence for each series obtained. What relation do the radii of convergence bear
to the point about which the series is obtained?
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b. Obtain the Laurent expansion about z = 2 of
f (z) = −
1+z
.
z(z − 2)
13. Classify the singularities (including the branch points) of the following functions
i)
14.
√
1+z
e1/z
1
, ii) 2
, iii)
,
and
iv)
z 2 − z − 2.
z(2 − z)
z +1
1 + ez
a. Find the residues of
2
1
ez
, at z0 = 1, ii) z
at z0 = 0.
i)
z−1
e −1
b. Explain what is wrong with the following reasoning. Let
f (z) =
1 + ez 1
+ .
z2
z
f (z) has a pole at z = 0 and the residue at that point is the coefficient of 1/z, namely
1.
Compute the residue correctly.
c. If φ(z) has a simple zero at z = a show that 1/φ(z) has a simple pole at z = a and
that the residue at z = a is 1/φ′ (a).
15. By considering the integral
Z
eikz dz
z2 + 1
taken round a large semicircle with centre at z = 0, evaluate
Z ∞
cos kt dt
, k > 0.
2
−∞ t + 1
16. Find the poles of 1/(z 2 − 2z + 4) and evaluate
Z ∞
dx
.
2
−∞ x − 2x + 4
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17. Find the first two terms in the Laurent expansion of f (z) about z = 0 where
f (z) =
eiaz − eibz
.
z2
Hence show that f (z) has a simple pole at z = 0. By integrating round a semicircular
closed contour indented at the origin show that
Z ∞
cos ax − cos bx
dx
2
+ π(a − b) = 0.
x2
0
NdA
September 24, 2015
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