96
2 Variational Problems with Fixed Boundaries
A few examples given in the above section shows how to deduce the functional of
the problem from a physical problem or geometric problem and the corresponding
physical laws or the geometric theorem. But only functional is not enough to solve the
variational problem in the study, some additional conditions should also be proposed
to restrict variational problem under study, these additional conditions to restrict the
variational problem is called the constraint. If give the additional condition that
an unknown function should satisfy at the endpoint of interval or on the boundary
of domain, then it is called the boundary condition of the variational problem. If
give the additional condition that an unknown function should satisfy at the initial
time, then it is called the initial condition of the variational problem. Other conditions are called the constraint conditions of the variational problem. The constraint
conditions can be divided into two types, the relation satisfied between the functional variables is called general constraint condition. When a functional attains an
extremum or stationary value, the condition obtained through variation operation the
functional is called the variational constraint condition or variational condition.
For instance, in Example 2.1.3, the isoperimetric condition given by Eq. (3) is a
kind of constraint condition. If an admissible function is not restricted on the border,
then the border is called the free boundary, the boundary condition is called the
free boundary condition. The initial condition, constraint condition and boundary
condition are collectively called the definite conditions or conditions for determining solution of the variational problem. A functional combined with certain
definite condition, which is called a definite problem or problem for determining
solution. In general, a definite problem probably has not the constraint condition,
but must have the boundary condition or the initial condition.
Let J [y(x)] be a functional defined in an admissible class function F = {y(x)},
y 0 (x) is a function in F. If for any y(x) in F, there is J = J [y(x)]− J [y 0 (x)] ≥ 0 or
≤ 0, then the functional J [y(x)] is called having the absolute minimum or absolute
maximum in y 0 (x), it is also called the global minimum or global maximum. The
corresponding y 0 (x) is called the having the absolute minimum function or absolute maximum function, it is also called the global minimum function or global
maximum function. The absolute minimum and absolute maximum are generally
called the absolute extremum or global extremum. The absolute minimum function and absolute maximum function are generally called the absolute extremum
function or global extremum function.
Example 2.2.3 Let 0 ≤ x 0 < x 1 , prove that the J [y(x)] =
x 1
x 0
(x
2
+ y
2
)dx has the
absolute minimum on the curve y(x) ≡ 0.
Proof Since 0 ≤ x 0 < x 1 , so an arbitrary continuous function y(x) in [x 0 , x 1 ], there
is
J = J [y(x)] − J [0] =
x 1
x 0
(x
2
+ y
2
)dx −
x 1
x 0
x
2 dx =
x 1
x 0
y
2 dx ≥ 0
The equality holds only when y(x) ≡ 0. Quod erat demonstrandum.
2 Variational Problems with Fixed Boundaries
A few examples given in the above section shows how to deduce the functional of
the problem from a physical problem or geometric problem and the corresponding
physical laws or the geometric theorem. But only functional is not enough to solve the
variational problem in the study, some additional conditions should also be proposed
to restrict variational problem under study, these additional conditions to restrict the
variational problem is called the constraint. If give the additional condition that
an unknown function should satisfy at the endpoint of interval or on the boundary
of domain, then it is called the boundary condition of the variational problem. If
give the additional condition that an unknown function should satisfy at the initial
time, then it is called the initial condition of the variational problem. Other conditions are called the constraint conditions of the variational problem. The constraint
conditions can be divided into two types, the relation satisfied between the functional variables is called general constraint condition. When a functional attains an
extremum or stationary value, the condition obtained through variation operation the
functional is called the variational constraint condition or variational condition.
For instance, in Example 2.1.3, the isoperimetric condition given by Eq. (3) is a
kind of constraint condition. If an admissible function is not restricted on the border,
then the border is called the free boundary, the boundary condition is called the
free boundary condition. The initial condition, constraint condition and boundary
condition are collectively called the definite conditions or conditions for determining solution of the variational problem. A functional combined with certain
definite condition, which is called a definite problem or problem for determining
solution. In general, a definite problem probably has not the constraint condition,
but must have the boundary condition or the initial condition.
Let J [y(x)] be a functional defined in an admissible class function F = {y(x)},
y 0 (x) is a function in F. If for any y(x) in F, there is J = J [y(x)]− J [y 0 (x)] ≥ 0 or
≤ 0, then the functional J [y(x)] is called having the absolute minimum or absolute
maximum in y 0 (x), it is also called the global minimum or global maximum. The
corresponding y 0 (x) is called the having the absolute minimum function or absolute maximum function, it is also called the global minimum function or global
maximum function. The absolute minimum and absolute maximum are generally
called the absolute extremum or global extremum. The absolute minimum function and absolute maximum function are generally called the absolute extremum
function or global extremum function.
Example 2.2.3 Let 0 ≤ x 0 < x 1 , prove that the J [y(x)] =
x 1
x 0
(x
2
+ y
2
)dx has the
absolute minimum on the curve y(x) ≡ 0.
Proof Since 0 ≤ x 0 < x 1 , so an arbitrary continuous function y(x) in [x 0 , x 1 ], there
is
J = J [y(x)] − J [0] =
x 1
x 0
(x
2
+ y
2
)dx −
x 1
x 0
x
2 dx =
x 1
x 0
y
2 dx ≥ 0
The equality holds only when y(x) ≡ 0. Quod erat demonstrandum.