 The Shortest
Path  A “Standard”
Derivation 
This is a
“classic” derivation of the minimization condition for a path,
using integration by parts. It's similar to proofs which appear in
any number of mechanics texts. (See, also, the
visual derivation
of the minimization condition.)
Integration
by parts is pretty simple in principle, but in practice I
typically am left feeling like the magician just pulled a rabbit from
a hat after its application. My preference would be for a
purely "visual" derivation. But let us proceed.
The
definition of a path and statement of the problem are the same as those
given with the
visual
derivation. If you've already seen them
there, you might as well just
skip
down to the
derivation.
Definition of a Path
We define
a
path in R
^{N} from
point X
_{a }to point X
_{b} as
a smooth mapping from
the unit interval on the real line [0,1] into R
^{N}:
(1)
Figure 1

Some possible paths in R^{2}:
Statement of the Problem
We are
given a function
f which is
defined along any path. Along any
particular path,
f
is a function of the components
x^{i} of the path
X,
and
of their derivatives, which we show with an overdot.
f maps each point on the path into R. So,we could also say that
f
maps a particular path, and a point in the unit interval, into R; thus, we have:
(2)
We wish to find a particular path which
will minimize (or extremize) the integral of
f
over the
path. That integral we will call
F; it maps each
path into a point in R:
(3)
More specifically, we wish to find a path such that
the integral of
f along any
nearby
path is at least as large as the integral along our chosen path.
Figure
2  Some "nearby" paths in R^{2}:
Note, however, that
something important isn't shown on the illustrations: The
function,
f, is a function of
the location along the path, and is also a function of
how
fast we are moving along the path. The derivatives
of
x and
y with respect to
t
do not appear in figures 1 and 2 but they are important none the less.
The
condition a path must satisfy, to be a minimum (or maximum) of
F,
is the familiar one: The derivative of
F (with
respect to the
path) must vanish. That means that any
“infinitesimal”
deviation from the path will result in no change  or, in other
words, the path must be a stationary point for
F in the space of all
possible paths. We wish to find purely local conditions on
the
functon
X(t) which will allow us to determine if a path is minimal.
At
this point we have our definitions in place, and it would make sense
to bail out, look at the
visual
derivation of the minimization
conditions, and call it a day. But what follows is more rigorous.
The Derivation of the Minimality
Conditions
We
need to define a variation from a path. That's just another path,
but it's a path leading from
0
to
0. We'll call it
η:
(4)
Now, we want to take the
derivative
in the direction of a particular variation,
η. We
do that by
multiplying
η by a scalar, and then differentiating
with respect to
the scalar:
(5)
Rewriting that in terms of
f,
the condition we need to satisfy is:
(6)
Since we're only interested in the
derivative at s=0, we can replace the integrand with the first order
terms of its Taylor series:
(7)
The derivative now becomes:
(8)
We'd like to eliminate references to dη /dt from this, so we
integrate the second term by parts:
(9)
Because
η(0) =
η(1) =
0, the bracketed term in (9) disappears, and we're
left with:
(10)
Since this must be zero for any
arbitrary
η, we must, therefore, have:
(11)

For the path to be extremal, that is
the condition which must be satisfied.
Go
to the visual
derivation of the condition.
Page created
10/31/06; updated with prettier equations on 11/14/06