Physics 5040
Spring 2009
Problem Set 2
1. Recall in our derivation of the geodesic equation that extremizing
Z
Z
ds
dt
ds =
dt
is equivalent to extremizing
Z ds
dt
2
dt.
Show that the Euler-Lagrange equations for a geodesic in plane polar coordinates are the same as the coupled equations that were derived from the
general geodesic equation in Example 4 of the calculus of variations notes.
2. Using plane polar coordinates, show that a geodesic in the plane is a straight
line.
R√
2 θ ′2 dr (where
1 + r√
Hint : Show that the function to be minimized is I =
R
′
θ = dθ/dr), which is independent of θ. The integral c/r r2 − c2 dr is
easily done with the substitution r = c/ cos u, yielding an equation for r as a
function of θ − θ0 where θ0 is a constant of integration. Now show that this
is the equation of a straight line perpendicular to the direction θ0 .
3. Fermat’s principle states: If the velocity of light is given by the continuous
function u = u(y), the actual light path connecting the points (x1 , y1 ) and
(x2 , y2 ) in a plane is one which extremizes the time integral
Z x2 ,y2 )
ds
I=
.
(x1 ,y1 ) u
Derive Snell’s law from Fermat’s principle; that is, prove that sin θ/u = const,
where θ is the angle shown below.
y
θ
•
(x2 , y2 )
y(x)
•
(x1 , y1 )
x
1
Hint : Use the second form of Euler’s equation (equation 14 in the notes).
4. We have seen that the geodesic equation is
j
i
d2 xl
l dx dx
+
Γ
= 0.
ij
dt2
dt dt
Show that under a change of parameter t → σ = f (t) the geodesic equation
becomes
j
i
f¨ dxl
d2 xl
l dx dx
+
Γ
=
−
.
ij
dσ 2
dσ dσ
f˙2 dσ
Thus the geodesic equation retains its form only under affine changes of parameter t → f (t) = at + b.
5. Show that the geodesic equation implies that the quantity
gij
dxi dxj
dt dt
is a constant along the geodesic, i.e.,
dxi dxj
d
gij
= 0.
dt
dt dt
Hint : Don’t forget to use the geodesic equation. Also recall the formula for
the Christoffel symbols
∂gki
∂gij
1 lk ∂gkj
l
.
+
−
Γij = g
2
∂xi
∂xj
∂xk
6. Using the geodesic equation, show that the geodesics on the surface of a sphere
in R3 are great circles.
7. (a) Using the notation of Section 7 of the calculus of variations notes, suppose
that we want to extremize the functional
Z t1
˙ x̃,
¨ t) dt
f (x̃, x̃,
I(ε) =
t0
where all other conditions (such as fixed end points and the continuity of
derivatives) are as in the notes. Derive the Euler-Lagrange equation for
f . Be sure to clearly state any additional boundary conditions you need
to specify in order to obtain your result.
(b) Find the equation of motion of a particle described by the Lagrangian
L=−
What motion does this describe?
2
k
m
q q̈ − q 2 .
2
2
8. The parametrized equation of a circular helix of radius ρ in R3 is of the form
x(t) = (ρ cos ωt, ρ sin ωt, kt).
Show that the geodesics on the surface of a right circular cylinder of radius ρ
oriented along the z-axis in R3 are portions of a helix.
9. Let F [y] be the functional
F [y] =
Z
sin y(x)
dx .
y(x)
Using the definition
F [y(x) + εδ(x − x0 )] − F [y]
δF [y]
= lim
δy(x0 ) ε→0
ε
find
δF [y]
.
δy(x0 )
10. (a) In Example 10 of the notes it is shown that as a consequence of Green’s
theorem, the area enclosed by a simple closed curve is given by
Z
1
A=
(xẏ − ẋy) dt.
2
Show by changing to plane polar coordinates that this is the same as
Z
1
r2 dθ.
A=
2
(b) Inverse Isoperimetric Problem (“isoareametric” problem). Prove that of
all simple closed curves enclosing a given area A, the least perimeter is
possessed by the circle. What is the value of the Lagrange multiplier λ in
terms of A?
11. Suppose you have a string of length l with one end fixed at the origin of the
xy-plane, and the other end on the x-axis such that the area enclosed between
the string and the x-axis is a maximum. Show that the shape of the string is
a semicircle.
R
Hint : From elementary calculus the area is given by y dx. Show that this is
Rl p
the same as 0 y 1 − y ′2 ds where s is the distance along the string starting
at the origin, and y ′ = dy/ds. Use the second form of Euler’s equation to
find y as a function of s. Now use this to find x as a function of s, and then
combine these equations for x and y to eliminate s.
12. Find the dimensions of the parallelepiped of maximum volume inscribed in a
sphere of radius R.
3