Editing Calculus of variations (section)

== Variations and sufficient condition for a minimum ==

Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument.  The '''first variation'''{{efn|name=AltFirst| The first variation is also called the variation, differential, or first differential.}} is defined as the linear part of the change in the functional, and the '''second variation'''{{efn|name=AltSecond| The second variation is also called the second differential.}} is defined as the quadratic part.<ref name='GelfandFominP11–12,99'>{{harvnb|Gelfand|Fomin|2000|pp=11–12, 99}}</ref>

For example, if <math>J[y]</math> is a functional with the function <math>y = y(x)</math> as its argument, and there is a small change in its argument from <math>y</math> to <math>y + h,</math> where <math>h = h(x)</math> is a function in the same function space as <math>y,</math> then the corresponding change in the functional is{{efn|name=SimplifyNotation|Note that <math>\Delta J[h]</math> and the variations below, depend on both <math>y</math> and <math>h.</math> The argument <math>y</math> has been left out to simplify the notation. For example, <math>\Delta J[h]</math> could have been written <math>\Delta J[y; h].</math><ref name='GelfandFominP12FN6'>{{harvnb | Gelfand|Fomin|2000 | p=12, footnote 6}}</ref>}}
<math display="block">\Delta J[h] = J[y+h] - J[y].</math>

The functional <math>J[y]</math> is said to be '''differentiable''' if
<math display="block">\Delta J[h] = \varphi [h] + \varepsilon \|h\|,</math>
where <math>\varphi[h]</math> is a linear functional,{{efn|name=Linear|A functional <math>\varphi[h]</math> is said to be '''linear''' if <math>\varphi[\alpha h] = \alpha \varphi[h]</math> &nbsp; and &nbsp; <math>\varphi\left[h + h_2\right] = \varphi[h] + \varphi\left[h_2\right],</math> where <math>h, h_2</math> are functions and <math>\alpha</math> is a real number.<ref name='GelfandFominP8'>{{harvnb | Gelfand|Fomin| 2000 | p=8 }}</ref>}} <math>\|h\|</math> is the norm of <math>h,</math>{{efn|name=Norm| For a function <math>h = h(x)</math> that is defined for <math>a \leq x \leq b,</math> where <math>a</math> and <math>b</math> are real numbers, the norm of <math>h</math> is its maximum absolute value, i.e. <math>\|h\| = \displaystyle\max_{a \leq x \leq b} |h(x)|.</math><ref name='GelfandFominP6'>{{harvnb | Gelfand|Fomin| 2000 | p=6 }}</ref>}} and <math>\varepsilon \to 0</math> as <math>\|h\| \to 0.</math> The linear functional <math>\varphi[h]</math> is the first variation of <math>J[y]</math> and is denoted by,<ref name='GelfandFominP11–12'>{{harvnb | Gelfand|Fomin| 2000 | pp=11–12}}</ref>
<math display="block">\delta J[h] = \varphi[h].</math>

The functional <math>J[y]</math> is said to be '''twice differentiable''' if
<math display="block">\Delta J[h] = \varphi_1 [h] + \varphi_2 [h] + \varepsilon \|h\|^2,</math>
where <math>\varphi_1[h]</math> is a linear functional (the first variation), <math>\varphi_2[h]</math> is a quadratic functional,{{efn|name=Quadratic| A functional is said to be '''quadratic''' if it is a bilinear functional with two argument functions that are equal. A '''bilinear functional''' is a functional that depends on two argument functions and is linear when each argument function in turn is fixed while the other argument function is variable.<ref name='GelfandFominP97–98'>{{harvnb | Gelfand|Fomin| 2000 | pp=97–98 }}</ref>}} and <math>\varepsilon \to 0</math> as <math>\|h\| \to 0.</math> The quadratic functional <math>\varphi_2[h]</math> is the second variation of <math>J[y]</math> and is denoted by,<ref name='GelfandFominP99'>{{harvnb | Gelfand|Fomin| 2000 | p=99 }}</ref>
<math display="block">\delta^2 J[h] = \varphi_2[h].</math>

The second variation <math>\delta^2 J[h]</math> is said to be '''strongly positive''' if
<math display="block">\delta^2J[h] \ge k \|h\|^2,</math>
for all <math>h</math> and for some constant <math>k > 0</math>.<ref name='GelfandFominP100'>{{harvnb | Gelfand|Fomin| 2000 | p=100 }}</ref>

Using the above definitions, especially the definitions of first variation, second variation, and strongly positive, the following sufficient condition for a minimum of a functional can be stated.

{{quote box|align=left |fontsize=100% |border=2px |quote='''Sufficient condition for a minimum:'''

{{block indent | em = 1.5 | text = The functional <math>J[y]</math> has a minimum at <math>y = \hat{y}</math> if its first variation <math>\delta J[h] = 0</math> at <math>y = \hat{y}</math> and its second variation <math>\delta^2 J[h]</math> is strongly positive at <math>y = \hat{y}.</math><ref name='GelfandFominP100Theorem2'>{{harvnb | Gelfand|Fomin| 2000 | p=100 |loc=Theorem 2}}</ref>{{Efn|name=sufficient| For other sufficient conditions, see in {{harvnb|Gelfand|Fomin|2000}},
* '''Chapter{{nbsp}}5: "The Second Variation. Sufficient Conditions for a Weak Extremum" – ''' Sufficient conditions for a weak minimum are given by the theorem on p.{{nbsp}}116.
* '''Chapter{{nbsp}}6: "Fields. Sufficient Conditions for a Strong Extremum" – ''' Sufficient conditions for a strong minimum are given by the theorem on p.{{nbsp}}148.}}{{efn|name=FuncMin| One may note the similarity to the sufficient condition for a minimum of a function, where the first derivative is zero and the second derivative is positive.}} }}}}
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