2.2 Fundamental Conceptions of the Calculus of Variations
97
If the function that is as a comparison is only restricted to a neighborhood of
y 0 (x), with J = J [y(x)] − J [y 0 (x)] ≥ 0 or ≤ 0, then the functional J [y(x)] is
called having the relative minimum or relative maximum in y 0 (x), is also called
the local minimum or local maximum. The local minimum or local maximum are
generally called relative extremum or local extremum.
The extremum of a functional is associated with the approach degree of the functional variables. Let J [y(x)] be the functional defined in an admissible class function
F = {y(x)}, y 0 (x) is a function in F. If in the zero-th order δ neighborhood of y 0 (x),
there is J = J [y(x)] − J [y 0 (x)] ≥ 0 (or ≤ 0), then the functional J [y(x)]
is called having the strong relative minimum (or strong relative maximum) or
strong minimum (or strong maximum) in y 0 (x). If in the first order δ neighborhood of y 0 (x), there is J = J [y(x)] − J [y 0 (x)] ≥ 0 (or ≤ 0), then the functional
J [y(x)] is called having the weak relative minimum (or weak relative maximum)
or weak minimum (or weak maximum) in y 0 (x). The strong (weak) minimum
and strong (weak) maximum are generally called the strong (weak) extremum. The
distinction between a strong extremum and weak extremum value in discussing the
necessary condition of extremum of a functional play an unimportant role, iut in the
research on the sufficient condition of extremum of a functional, this distinction is
very important.
Each absolute extremum meanwhile is also strong relative extremum or weak
relative extremum, however, in turn, each relative extremum is not necessarily an
absolute extremum.
The absolute extremum, relative minimum, strong extremum and weak extremum
are generally called the extremum (value), extremal (value) or extreme (value).
Because the first order neighborhood is a part of the zero-th order neighborhood,
and the two functions with the first order approach degree certainly have the zero-th
order approach degree, so if the functional J [y(x)] has a strong extremum in the
function y 0 (x), then it must also have a weak extremum in y 0 (x). But otherwise
it is not quite so, namely if the functional J [y(x)] has a weak extremum in the
function y 0 (x), but may not have a strong extremum in the function y 0 (x). That is
to say, for those curves y(x) near the curve y 0 (x) on both the ordinate and y
0 (x)
(namely on the tangential direction), there may be not such a curve that J [y(x)] <
J [y 0 (x)] (or J [y(x)] > J [y 0 (x)]), however for those curves y(x) near the curve
y 0 (x) only on the ordinate but not necessary on y
0 (x), then may find such a curve
that J [y(x)] < J [y 0 (x)] (or J [y(x)] > J [y 0 (x)]). Therefore, for the necessary
condition of weak relative extremum of the functional J [y(x)], which must be the
necessary conditions of its strong relative extremum. As a result, when discussing
the necessary condition of extremum of a functional, always let δ neighborhood be
the first order neighborhood.
It can be known from the definition of the absolute extremum, strong extremum
and weak extremum that the domain of the strong extremum is a part of the domain
of the absolute extremum, moreover the domain of the weak extremum is a part of
the domain of the strong extremum, therefore, a functional can has a value within
the large scale domain, it certainly has a value within the small scale domain, but put
the other way round, it is not necessarily true.
97
If the function that is as a comparison is only restricted to a neighborhood of
y 0 (x), with J = J [y(x)] − J [y 0 (x)] ≥ 0 or ≤ 0, then the functional J [y(x)] is
called having the relative minimum or relative maximum in y 0 (x), is also called
the local minimum or local maximum. The local minimum or local maximum are
generally called relative extremum or local extremum.
The extremum of a functional is associated with the approach degree of the functional variables. Let J [y(x)] be the functional defined in an admissible class function
F = {y(x)}, y 0 (x) is a function in F. If in the zero-th order δ neighborhood of y 0 (x),
there is J = J [y(x)] − J [y 0 (x)] ≥ 0 (or ≤ 0), then the functional J [y(x)]
is called having the strong relative minimum (or strong relative maximum) or
strong minimum (or strong maximum) in y 0 (x). If in the first order δ neighborhood of y 0 (x), there is J = J [y(x)] − J [y 0 (x)] ≥ 0 (or ≤ 0), then the functional
J [y(x)] is called having the weak relative minimum (or weak relative maximum)
or weak minimum (or weak maximum) in y 0 (x). The strong (weak) minimum
and strong (weak) maximum are generally called the strong (weak) extremum. The
distinction between a strong extremum and weak extremum value in discussing the
necessary condition of extremum of a functional play an unimportant role, iut in the
research on the sufficient condition of extremum of a functional, this distinction is
very important.
Each absolute extremum meanwhile is also strong relative extremum or weak
relative extremum, however, in turn, each relative extremum is not necessarily an
absolute extremum.
The absolute extremum, relative minimum, strong extremum and weak extremum
are generally called the extremum (value), extremal (value) or extreme (value).
Because the first order neighborhood is a part of the zero-th order neighborhood,
and the two functions with the first order approach degree certainly have the zero-th
order approach degree, so if the functional J [y(x)] has a strong extremum in the
function y 0 (x), then it must also have a weak extremum in y 0 (x). But otherwise
it is not quite so, namely if the functional J [y(x)] has a weak extremum in the
function y 0 (x), but may not have a strong extremum in the function y 0 (x). That is
to say, for those curves y(x) near the curve y 0 (x) on both the ordinate and y
0 (x)
(namely on the tangential direction), there may be not such a curve that J [y(x)] <
J [y 0 (x)] (or J [y(x)] > J [y 0 (x)]), however for those curves y(x) near the curve
y 0 (x) only on the ordinate but not necessary on y
0 (x), then may find such a curve
that J [y(x)] < J [y 0 (x)] (or J [y(x)] > J [y 0 (x)]). Therefore, for the necessary
condition of weak relative extremum of the functional J [y(x)], which must be the
necessary conditions of its strong relative extremum. As a result, when discussing
the necessary condition of extremum of a functional, always let δ neighborhood be
the first order neighborhood.
It can be known from the definition of the absolute extremum, strong extremum
and weak extremum that the domain of the strong extremum is a part of the domain
of the absolute extremum, moreover the domain of the weak extremum is a part of
the domain of the strong extremum, therefore, a functional can has a value within
the large scale domain, it certainly has a value within the small scale domain, but put
the other way round, it is not necessarily true.