2.1 Examples of the Classical Variational Problems
89
Example 2.1.3 The isoperimetric problem. In all disjoint smooth closed curves of
a given length L in the plane, find out a curve that can round into the largest area. This
is the origin of the name. This is one of the oldest variational problem, as early as
in ancient Greece, people would know that this curve was a circle. the problem was
solved by Jacob Bernoulli in 1701, but its variational characteristics was not solved by
Euler until 1744, but there are also people who believe that in 1732 when expounding
the general solution of the isoperimetric problem, Euler gave the general methods of
solving variational problem. Generally speaking, the problem making a functional
obtain extremum and at the same time making another functional be given value is
called the isoperimetric problem. In other words, in the constraint condition that
the value of a functional is given, another functional obtains extremum, this problem
is called the isoperimetric problem.
Solution Let the parameter equations of a closed curve be
x = x(t), y = y(t) t 0 ≤ t ≤ t 1
(1)
where, the functions x(t), y(t) are continuously differentiable, and x(t 0 ) = x(t 1 ),
y(t 0 ) = y(t 1 ), t 0 and t 1 correspond to the starting point and end point of the closed
curve.
Moreover suppose that the length of the closed curve is L, namely
L =
(dx) 2 + (dy) 2 =
t 1
t 0
˙
x 2 (t) + ˙
y 2 (t)dt
(2)
According to the Green’s theorem, the area surrounded by the curve is
A =
1
2
(xdy − ydx) =
1
2
(x ˙
y − y ˙
x)dt
(3)
Thus, the isoperimetric problem is, in all of the curves (1) which satisfy the
condition (2), to find the curve that makes the integral (3) have the maximum value.
Equation (2) is called the isoperimetric condition.
The three classical variational examples in history had a close relationship and a
strong impact on the development of variational methods, they appeared at the end of
the 17th century and the early 18th century, and were the touchstone verifying a new
mathematical analysis method. It can be seen from the above mentioned examples
that the variables T, L and A to obtain extremum and given by the definite integral all
depend on the selection of unknown curves or unknown surfaces. These unknown
curves or surfaces are usually given by a number of functions, so these variables
given by the definite integral can be regarded as that they depend on the unknown
functions and the derivative of the unknown functions, and the unknown functions
and the derivative of the unknown functions at variable independently change, act as
arguments, such arguments called the independent function. Briefly, this function
that depends on the independent function is called the functional, or the function
89
Example 2.1.3 The isoperimetric problem. In all disjoint smooth closed curves of
a given length L in the plane, find out a curve that can round into the largest area. This
is the origin of the name. This is one of the oldest variational problem, as early as
in ancient Greece, people would know that this curve was a circle. the problem was
solved by Jacob Bernoulli in 1701, but its variational characteristics was not solved by
Euler until 1744, but there are also people who believe that in 1732 when expounding
the general solution of the isoperimetric problem, Euler gave the general methods of
solving variational problem. Generally speaking, the problem making a functional
obtain extremum and at the same time making another functional be given value is
called the isoperimetric problem. In other words, in the constraint condition that
the value of a functional is given, another functional obtains extremum, this problem
is called the isoperimetric problem.
Solution Let the parameter equations of a closed curve be
x = x(t), y = y(t) t 0 ≤ t ≤ t 1
(1)
where, the functions x(t), y(t) are continuously differentiable, and x(t 0 ) = x(t 1 ),
y(t 0 ) = y(t 1 ), t 0 and t 1 correspond to the starting point and end point of the closed
curve.
Moreover suppose that the length of the closed curve is L, namely
L =
(dx) 2 + (dy) 2 =
t 1
t 0
˙
x 2 (t) + ˙
y 2 (t)dt
(2)
According to the Green’s theorem, the area surrounded by the curve is
A =
1
2
(xdy − ydx) =
1
2
(x ˙
y − y ˙
x)dt
(3)
Thus, the isoperimetric problem is, in all of the curves (1) which satisfy the
condition (2), to find the curve that makes the integral (3) have the maximum value.
Equation (2) is called the isoperimetric condition.
The three classical variational examples in history had a close relationship and a
strong impact on the development of variational methods, they appeared at the end of
the 17th century and the early 18th century, and were the touchstone verifying a new
mathematical analysis method. It can be seen from the above mentioned examples
that the variables T, L and A to obtain extremum and given by the definite integral all
depend on the selection of unknown curves or unknown surfaces. These unknown
curves or surfaces are usually given by a number of functions, so these variables
given by the definite integral can be regarded as that they depend on the unknown
functions and the derivative of the unknown functions, and the unknown functions
and the derivative of the unknown functions at variable independently change, act as
arguments, such arguments called the independent function. Briefly, this function
that depends on the independent function is called the functional, or the function