90
2 Variational Problems with Fixed Boundaries
that takes a function as the argument is called the functional. The functional in the
integral form is called the integral type functional, functional of integral type,
functional with integral or integral functional. The general definition about the
functional is given in the next section.
The above mentioned three classical variational problems will be solved in
succession in later.
2.2 Fundamental Conceptions of the Calculus of Variations
In order to discuss the convenience of problems, here are some concepts related to
the variational methods.
A collection of functions that have some common properties is called the class
function, function class or class of functions, it is written as F. For instance, in
Example 2.1.1, All of the plane curves pass through point A and point B, therefore
every function passes through point A and point B is the common property of the
function set. In Example 2.1.3, the perimeters of all smooth closed curves are given
L, L is a common property of the function set.
The common class functions have:
The continuous function set in the open interval (x 0 , x 1 ), it is called the continuous
function class in an open interval (x 0 , x 1 ), it is written as C(x 0 , x 1 ).
The continuous function set in the closed interval [x 0 , x 1 ], it is called the continuous function class in a closed interval [x 0 , x 1 ], it is written as C[x 0 , x 1 ], where the
function at the left endpoint of the interval is right continuous, at the right endpoint
of the interval is left continuous. At this moment the continuity at an endpoint of the
interval is called one-sided continuity.
In an open interval (x 0 , x 1 ), the function set that n th derivative is continuous
is called the function class of continuous n th derivative in an open interval
(x 0 , x 1 ), it is written as F = { y(x)|y ∈ C
n
(x 0 , x 1 ) or C
n
(x 0 , x 1 ), and stipulate
C
0
(x 0 , x 1 ) = C(x 0 , x 1 ), namely the zero-order derivative of the function class is the
function class itself.
The symbol C
n
(x 0 , x 1 ) denotes that the function set in a neighborhood of each
point in the open interval (x 0 , x 1 ) is defined, in (x 0 , x 1 ) there is the set of all the
functions y(x) of ≤ n all-order continuous derivatives. If for each n, there is y(x) ∈
C
n
(x 0 , x 1 ), then y(x) is called the infinite differentiable function, it is written as
y(x) ∈ C
n
(x 0 , x 1 ) ≡
∞
∩
n=0
C
n
(x 0 , x 1 ).
In a closed interval [x 0 , x 1 ], the function set that n th derivative is continuous is
called the function class of continuous n th derivative in a closed interval [x 0 , x 1 ],
it is written as F = { y(x)|y ∈ C
n
[x 0 , x 1 ], y(x 0 ) = y 0 , y(x 1 ) = y 1 } or C
n
[x 0 , x 1 ],
where the n th derivatives of the function y are one-sided continuity at the endpoints
of the interval, y 0 and y 1 are fixed constants, and stipulate C
0
[x 0 , x 1 ] = C[x 0 , x 1 ].
In an open interval (x 0 , x 1 ) and closed interval [x 0 , x 1 ], the function set that n th
derivative is continuous can all be written as F = {y(x)}.
2 Variational Problems with Fixed Boundaries
that takes a function as the argument is called the functional. The functional in the
integral form is called the integral type functional, functional of integral type,
functional with integral or integral functional. The general definition about the
functional is given in the next section.
The above mentioned three classical variational problems will be solved in
succession in later.
2.2 Fundamental Conceptions of the Calculus of Variations
In order to discuss the convenience of problems, here are some concepts related to
the variational methods.
A collection of functions that have some common properties is called the class
function, function class or class of functions, it is written as F. For instance, in
Example 2.1.1, All of the plane curves pass through point A and point B, therefore
every function passes through point A and point B is the common property of the
function set. In Example 2.1.3, the perimeters of all smooth closed curves are given
L, L is a common property of the function set.
The common class functions have:
The continuous function set in the open interval (x 0 , x 1 ), it is called the continuous
function class in an open interval (x 0 , x 1 ), it is written as C(x 0 , x 1 ).
The continuous function set in the closed interval [x 0 , x 1 ], it is called the continuous function class in a closed interval [x 0 , x 1 ], it is written as C[x 0 , x 1 ], where the
function at the left endpoint of the interval is right continuous, at the right endpoint
of the interval is left continuous. At this moment the continuity at an endpoint of the
interval is called one-sided continuity.
In an open interval (x 0 , x 1 ), the function set that n th derivative is continuous
is called the function class of continuous n th derivative in an open interval
(x 0 , x 1 ), it is written as F = { y(x)|y ∈ C
n
(x 0 , x 1 ) or C
n
(x 0 , x 1 ), and stipulate
C
0
(x 0 , x 1 ) = C(x 0 , x 1 ), namely the zero-order derivative of the function class is the
function class itself.
The symbol C
n
(x 0 , x 1 ) denotes that the function set in a neighborhood of each
point in the open interval (x 0 , x 1 ) is defined, in (x 0 , x 1 ) there is the set of all the
functions y(x) of ≤ n all-order continuous derivatives. If for each n, there is y(x) ∈
C
n
(x 0 , x 1 ), then y(x) is called the infinite differentiable function, it is written as
y(x) ∈ C
n
(x 0 , x 1 ) ≡
∞
∩
n=0
C
n
(x 0 , x 1 ).
In a closed interval [x 0 , x 1 ], the function set that n th derivative is continuous is
called the function class of continuous n th derivative in a closed interval [x 0 , x 1 ],
it is written as F = { y(x)|y ∈ C
n
[x 0 , x 1 ], y(x 0 ) = y 0 , y(x 1 ) = y 1 } or C
n
[x 0 , x 1 ],
where the n th derivatives of the function y are one-sided continuity at the endpoints
of the interval, y 0 and y 1 are fixed constants, and stipulate C
0
[x 0 , x 1 ] = C[x 0 , x 1 ].
In an open interval (x 0 , x 1 ) and closed interval [x 0 , x 1 ], the function set that n th
derivative is continuous can all be written as F = {y(x)}.