2.2 Fundamental Conceptions of the Calculus of Variations
95
Example 2.2.1 Let the curve y(x) =
sin n
k x
n
, where, k ≥ 2, n is a sufficiently large
positive integer, in the interval [0, π], the curve y 1 (x) ≡ 0. Find the zero-th order
and first order distance of the two curves.
Solution The zero-th order distance of the two curves is
d 0 = |y(x) − y 1 (x)| =
sin n
k x
n
− 0
=
sin n
k x
n
≤
1
n
The first order distance of the two curves is
d 1 =
y
(x) − y
1 (x)
= n
k−1
cos n
k x
At point x =
π
n k , there is
y
(x) − y
1 (x)
= n
k−1 , thus, for a sufficiently large
positive integer n, the first order distance d 1 can be arbitrarily large. Any given δ > 0,
when n > 1/δ, the two curves have the zero-th order δ approach degree, but they
have not the first order δ approach degree.
Example 2.2.2 Find the zero-th order distance of the two given curves y(x) = x
and y 1 (x) = ln x in the interval [e
−1 , e].
Solution From the definition, the zero-th order distance between the two curves is
d 0 = max
e −1 ≤x≤e
|y(x) − y 1 (x)| = max
e −1 ≤x≤e
|x − ln x| or d 0 = max
e −1 ≤x≤e
(x − ln x)
Derive d 0 and make it equal to zero, find out x = 1, but the zero-th order distance
at x = 1 is less than the first order distance at the right endpoint, so the zero-th order
distance of the two curves is d 0 = e − 1.
If a function of a class function class can make a functional obtain extremum or
may obtain extremum, then the class function class is called the admissible class
function of a variational problem. Generally speaking, the functions in an admissible
class function can have infinite numbers, anyone of them is called the admissible
function. The admissible functions in an admissible class function does not mean a
fixed function relationship, but they can be arbitrarily chosen and change in admissible class function. The curve (or surface) of the corresponding admissible class
function is called the admissible curve (or surface) class (or family), class (or
family or variety) of admissible curves (or surfaces). In a class function, the function (or curve) that can make a functional obtain extremum or may obtain extremum
is called the extremal function, extremal curve or extremal, it is also called the
solution of a variational problem. The core problem of variational methods is to
solve the extremal function of a functional and the extremum of the functional of the
corresponding extremum function. If the curve endpoint of an admissible curve class
is given in advance and has the fixed value, then the problem to find the extremum
of a functional is called the fixed end point variational problem, fixed boundary
variational problem, variational problem with fixed endpoints or variational
problem with fixed ends.
95
Example 2.2.1 Let the curve y(x) =
sin n
k x
n
, where, k ≥ 2, n is a sufficiently large
positive integer, in the interval [0, π], the curve y 1 (x) ≡ 0. Find the zero-th order
and first order distance of the two curves.
Solution The zero-th order distance of the two curves is
d 0 = |y(x) − y 1 (x)| =
sin n
k x
n
− 0
=
sin n
k x
n
≤
1
n
The first order distance of the two curves is
d 1 =
y
(x) − y
1 (x)
= n
k−1
cos n
k x
At point x =
π
n k , there is
y
(x) − y
1 (x)
= n
k−1 , thus, for a sufficiently large
positive integer n, the first order distance d 1 can be arbitrarily large. Any given δ > 0,
when n > 1/δ, the two curves have the zero-th order δ approach degree, but they
have not the first order δ approach degree.
Example 2.2.2 Find the zero-th order distance of the two given curves y(x) = x
and y 1 (x) = ln x in the interval [e
−1 , e].
Solution From the definition, the zero-th order distance between the two curves is
d 0 = max
e −1 ≤x≤e
|y(x) − y 1 (x)| = max
e −1 ≤x≤e
|x − ln x| or d 0 = max
e −1 ≤x≤e
(x − ln x)
Derive d 0 and make it equal to zero, find out x = 1, but the zero-th order distance
at x = 1 is less than the first order distance at the right endpoint, so the zero-th order
distance of the two curves is d 0 = e − 1.
If a function of a class function class can make a functional obtain extremum or
may obtain extremum, then the class function class is called the admissible class
function of a variational problem. Generally speaking, the functions in an admissible
class function can have infinite numbers, anyone of them is called the admissible
function. The admissible functions in an admissible class function does not mean a
fixed function relationship, but they can be arbitrarily chosen and change in admissible class function. The curve (or surface) of the corresponding admissible class
function is called the admissible curve (or surface) class (or family), class (or
family or variety) of admissible curves (or surfaces). In a class function, the function (or curve) that can make a functional obtain extremum or may obtain extremum
is called the extremal function, extremal curve or extremal, it is also called the
solution of a variational problem. The core problem of variational methods is to
solve the extremal function of a functional and the extremum of the functional of the
corresponding extremum function. If the curve endpoint of an admissible curve class
is given in advance and has the fixed value, then the problem to find the extremum
of a functional is called the fixed end point variational problem, fixed boundary
variational problem, variational problem with fixed endpoints or variational
problem with fixed ends.