0Introduction
IB Variational Principles
0 Introduction
Consider a light ray travelling towards a mirror and being reflected.
z
We see that the light ray travels towards the mirror, gets reflected at
z
, and hits
the (invisible) eye. What determines the path taken? The usual answer would
be that the reflected angle shall be the same as the incident angle. However,
ancient Greek mathematician Hero of Alexandria provided a different answer:
the path of the light minimizes the total distance travelled.
We can assume that light travels in a straight line except when reflected.
Then we can characterize the path by a single variable
z
, the point where the
light ray hits the mirror. Then we let
L
(
z
) to be the length of the path, and we
can solve for z by setting L
0
(z) = 0.
This principle sounds reasonable - in the absence of mirrors, light travels in
a straight line - which is the shortest path between two points. But is it always
true that the shortest path is taken? No! We only considered a plane mirror, and
this doesn’t hold if we have, say, a spherical mirror. However, it turns out that
in all cases, the path is a stationary point of the length function, i.e. L
0
(z) = 0.
Fermat put this principle further. Assuming that light travels at a finite
speed, the shortest path is the path that takes the minimum time. Fermat’s
principle thus states that
Light travels on the path that takes the shortest time.
This alternative formulation has the advantage that it applies to refraction as
well. Light travels at different speeds in different mediums. Hence when they
travel between mediums, they will change direction such that the total time
taken is minimized.
We usually define the refractive index
n
of a medium to be
n
= 1
/v
, where
v
is the velocity of light in the medium. Then we can write the variational
principle as
minimize
Z
path
n ds,
where d
s
is the path length element. This is easy to solve if we have two distinct
mediums. Since light travels in a straight line in each medium, we can simply
characterize the path by the point where the light crosses the boundary. However,
in the general case, we should be considering any possible path between two
points. In this case, we could no longer use ordinary calculus, and need new
tools - calculus of variations.
In calculus of variations, the main objective is to find a function
x
(
t
) that
minimizes an integral
R
f
(
x
) d
t
for some function
f
. For example, we might
want to minimize
R
(
x
2
+
x
) d
t
. This differs greatly from ordinary minimization
problems. In ordinary calculus, we minimize a function
f
(
x
) for all possible
values of
x ∈ R
. However, in calculus of variations, we will be minimizing the
integral
R
f(x) dt over all possible functions x(t).