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Suppose that we wish to find the function which
maximizes or minimizes the functional

(691) 
subject to the constraint that the value of

(692) 
remains constant. We can achieve our goal by finding an extremum of the new functional
, where is an undetermined function. We know
that , since the value of is fixed, so if then
as well. In other words, finding an extremum of is equivalent
to finding an extremum of . Application of the EulerLagrange
equation yields

(693) 
In principle, the above equation, together with the constraint (692),
yields the functions and . Incidentally, is generally
termed a Lagrange multiplier. If and have no explicit dependence then is usually a constant.
As an example, consider the following famous problem. Suppose that a uniform
chain of fixed length is suspended by its ends from
two equalheight fixed points which are a distance apart, where .
What is the equilibrium configuration of the chain?
Suppose that the chain has the uniform density per unit length .
Let the  and axes be horizontal and vertical, respectively, and
let the two ends of the chain lie at . The equilibrium configuration of the chain is specified by the function , for
, where
is the vertical distance of the chain below its end points at horizontal
position . Of course,
.
According to the discussion in Section 3.2, the stable equilibrium
state of a conservative dynamical system is one which minimizes
the system's potential energy. Now, the potential energy of the chain
is written

(694) 
where
is an element of length along the chain, and
is the acceleration due to gravity.
Hence, we need to minimize with respect to small variations in .
However, the variations in must be such as to conserve the
fixed length of the chain. Hence, our minimization procedure is subject to
the constraint that

(695) 
remains constant.
It follows, from the above discussion, that we need to minimize the
functional

(696) 
where is an, as yet, undetermined constant. Since the integrand
in the functional does not depend explicitly on , we have
from Equation (688) that

(697) 
where is a constant. This expression reduces to

(698) 
where
, and .
Let

(699) 
Making this substitution, Equation (698) yields

(700) 
Hence,

(701) 
where is a constant. It follows from Equation (699) that

(702) 
The above solution contains three undetermined constants, , , and . We can
eliminate two of these constants by application of the boundary
conditions . This yields

(703) 
Hence, , and
. It follows that

(704) 
The final unknown constant, , is determined via the application of
the constraint (695). Thus,

(705) 
Hence, the equilibrium configuration of the chain is given by the curve
(704), which is known as a catenary, where the parameter satisfies

(706) 
Next: MultiFunction Variation
Up: Hamiltonian Dynamics
Previous: Calculus of Variations
Richard Fitzpatrick
20110331