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Calculus of Variations
It is a wellknown fact, first enunciated by Archimedes, that the shortest
distance between two points in a plane is a straightline. However, suppose that
we wish to demonstrate this result from first principles. Let us consider the
length, , of various curves, , which run between two fixed
points, and , in a plane, as illustrated in Figure 35. Now, takes the form

(675) 
where
. Note that is a function of the function .
In mathematics, a function of a function is termed a functional.
Figure 35:
Different paths between points and .

Now, in order to find the shortest path between points and , we need to minimize the functional with respect to small variations
in the function , subject to the constraint that the end points,
and , remain fixed. In other words, we need to solve

(676) 
The meaning of the above equation is that if
, where is small, then the firstorder variation in ,
denoted ,
vanishes. In other words,
. The particular function
for which obviously yields an extremum of (i.e., either a maximum or a minimum). Hopefully,
in the case under consideration,
it yields a minimum of .
Consider a general functional of the form

(677) 
where the end points of the integration are fixed.
Suppose that
. The firstorder variation in is written

(678) 
where
. Setting to zero, we
obtain

(679) 
This equation must be satisfied for all possible small perturbations .
Integrating the second term in the integrand of the above equation by
parts, we get

(680) 
Now, if the end points are fixed then at
and . Hence, the last term on the lefthand side of the
above equation is zero. Thus, we obtain

(681) 
The above equation must be satisfied for all small perturbations
. The only way in which this is possible is for the
expression enclosed in square brackets in the integral to be zero. Hence, the functional
attains an extremum value whenever

(682) 
This condition is known as the EulerLagrange equation.
Let us consider some special cases. Suppose that does not explicitly
depend on . It follows that
. Hence,
the EulerLagrange equation (682) simplifies to

(683) 
Next, suppose that does not depend explicitly on . Multiplying
Equation (682) by , we obtain

(684) 
However,

(685) 
Thus, we get

(686) 
Now, if is not an explicit function of then the righthand side of
the above equation is the total derivative of , namely .
Hence, we obtain

(687) 
which yields

(688) 
Returning to the case under consideration, we have
, according to Equation (675) and (677). Hence, is not
an explicit function of , so Equation (683) yields

(689) 
where is a constant. So,

(690) 
Of course,
is the equation of a straightline. Thus, the shortest distance between two fixed points in a plane is indeed a
straightline.
Next: Conditional Variation
Up: Hamiltonian Dynamics
Previous: Introduction
Richard Fitzpatrick
20110331