2Euler-Lagrange equation
IB Variational Principles
2.3 Constrained variation of functionals
So far, we’ve considered the problem of finding stationary values of
F
[
x
] without
any restraint on what
x
could be. However, sometimes there might be some
restrictions on the possible values of
x
. For example, we might have a surface in
R
3
defined by
g
(
x
) = 0. If we want to find the path of shortest length on the
surface (i.e. geodesics), then we want to minimize
F
[
x
] subject to the constraint
g(x(t)) = 0.
We can again use Lagrange multipliers. The problem we have to solve is
equivalent to finding stationary values (without constraints) of
Φ
λ
[x] = F [x] − λ(P [x] − c).
with respect to the function x(t) and the variable λ.
Example (Isoperimetric problem). If we have a string of fixed length, what is
the maximum area we can enclose with it?
We first argue that the region enclosed by the curve is convex. If it is not,
we can “push out” the curve to increase the area enclosed without changing the
length. Assuming this, we can split the curve into two parts:
y
2
y
1
α
β
x
y
We have dA = [y
2
(x) − y
1
(x)] dx. So
A =
Z
β
α
[y
2
(x) − y
1
(x)] dx.
Alternatively,
A[y] =
I
y(x) dx.
and the length is
L[y] =
I
d` =
I
p
1 + (y
0
)
2
dx.
So we look for stationary points of
Φ
λ
[y] =
I
[y(x) − λ
p
1 + (y
0
)
2
] dx + λL.
In this case, we can be sure that our boundary terms vanish since there is no
boundary.
Since there is no explicit dependence on x, we obtain the first integral
f − y
0
∂f
∂y
0
= constant = y
0
.
So
y
0
= y −λ
p
1 + (y
0
)
2
+
λ(y
0
)
2
p
1 + (y
0
)
2
= y −
λ
p
1 + (y
0
)
2
.
So
(y −y
0
)
2
=
λ
2
1 + (y
0
)
2
(y
0
)
2
=
λ
2
(y −y
0
)
2
− 1
(y −y
0
)y
0
p
λ
2
− (y −y
0
)
2
= ±1.
d
h
p
λ
2
− (y −y
0
)
2
± x
i
= 0.
So we have
λ
2
− (y −y
0
)
2
= (x − x
0
)
2
,
or
(x − x
0
)
2
+ (y −y
0
)
2
= λ
2
.
This is a circle of radius
λ
. Since the perimeter of this circle will be 2
πλ
, we
must have λ = L/(2π). So the maximum area is πλ
2
= L
2
/(4π).
Example
(Sturm-Liouville problem)
.
The Sturm-Liouville problem is a very
general class of problems. We will develop some very general theory about these
problems without going into specific examples. It can be formulated as follows:
let
ρ
(
x
),
σ
(
x
) and
w
(
x
) be real functions of
x
defined on
α ≤ x ≤ β
. We will
consider the special case where
ρ
and
w
are positive on
α < x < β
. Our objective
is to find stationary points of the functional
F [y] =
Z
β
α
(ρ(x)(y
0
)
2
+ σ(x)y
2
) dx
subject to the condition
G[y] =
Z
β
α
w(x)y
2
dx = 1.
Using the Euler-Lagrange equation, the functional derivatives of F and G are
δF [y]
δy
= 2
− (ρy
0
)
0
+ σy
δG[y]
δy
= 2(wy).
So the Euler-Lagrange equation of Φ
λ
[y] = F [y] − λ(G[y] − 1) is
−(ρy
0
)
0
+ σy −λwy = 0.
We can write this as the eigenvalue problem
Ly = λwy.
where
L = −
d
dx
ρ
d
dx
+ σ
is the Sturm-Liouville operator. We call this a Sturm-Liouville eigenvalue
problem. w is called the weight function.
We can view this problem in a different way. Notice that
Ly
=
λwy
is linear
in
y
. Hence if
y
is a solution, then so is
Ay
. But if
G
[
y
] = 1, then
G
[
Ay
] =
A
2
.
Hence the condition
G
[
y
] = 1 is simply a normalization condition. We can get
around this problem by asking for the minimum of the functional
Λ[y] =
F [y]
G[y]
instead. It turns out that this Λ has some significance. To minimize Λ, we
cannot apply the Euler-Lagrange equations, since Λ is not of the form of an
integral. However, we can try to vary it directly:
δΛ =
1
G
δF −
F
G
2
δG =
1
G
(δF − ΛδG).
When Λ is minimized, we have
δΛ = 0 ⇔
δF
δy
= Λ
δG
δy
⇔ Ly = Λwy.
So at stationary values of Λ[y], Λ is the associated Sturm-Liouville eigenvalue.
Example
(Geodesics)
.
Suppose that we have a surface in
R
3
defined by
g
(
x
) = 0,
and we want to find the path of shortest distance between two points on the
surface. These paths are known as geodesics.
One possible approach is to solve
g
(
x
) = 0 directly. For example, if we have
a unit sphere, a possible solution is
x
=
cos θ cos φ
,
y
=
cos θ sin φ
,
z
=
sin θ
.
Then the total length of a path would be given by
D[θ, φ] =
Z
B
A
q
dθ
2
+ sin
2
θdφ
2
.
We then vary θ and φ to minimize D and obtain a geodesic.
Alternatively, we can impose the condition
g
(
x
(
t
)) = 0 with a Lagrange
multiplier. However, since we want the constraint to be satisfied for all
t
, we
need a Lagrange multiplier function
λ
(
t
). Then our problem would be to find
stationary values of
Φ[x, λ] =
Z
1
0
|
˙
x| − λ(t)g(x(t))
dt