View source for Fundamental lemma of calculus of variations

{{short description|Initial result in using test functions to find extremum}}
In [[Mathematics|mathematics]], specifically in the [[Calculus of variations|calculus of variations]], a variation {{math|''&delta;f''}} of a function {{math|''f''}} can be concentrated on an arbitrarily small interval, but not a single point.
Accordingly, the necessary condition of extremum ([[Functional derivative|functional derivative]] equal zero) appears in a [[Weak formulation|weak formulation]] (variational form) integrated with an arbitrary function {{math|''&delta;f''}}. The '''fundamental lemma of the calculus of variations''' is typically used to transform this weak formulation into the strong formulation ([[Differential equation|differential equation]]), free of the integration with arbitrary function. The proof usually exploits the possibility to choose {{math|''&delta;f''}} concentrated on an interval on which {{math|''f''}} keeps sign (positive or negative). Several versions of the lemma are in use. Basic versions are easy to formulate and prove. More powerful versions are used when needed.

==Basic version==

:If a continuous function <math>f</math> on an open interval <math>(a,b)</math> satisfies the equality
::<math> \int_a^b f(x)h(x)\,\mathrm{d}x = 0 </math>
:for all compactly supported smooth functions <math>h</math> on <math>(a,b)</math>, then <math>f</math> is identically zero.<ref name=J1>{{harvnb|Jost|Li-Jost|1998|loc=Lemma 1.1.1 on p.6}}</ref><ref name=GF1>{{harvnb|Gelfand|Fomin|1963|loc=Lemma 1 on p.9 (and Remark)}}</ref>

Here "smooth" may be interpreted as "infinitely differentiable",<ref name=J1 /> but often is interpreted as "twice continuously differentiable" or "continuously differentiable" or even just "continuous",<ref name=GF1 /> since these weaker statements may be strong enough for a given task. "Compactly supported" means "vanishes outside <math>[c,d]</math> for some <math>c</math>, <math>d</math> such that <math>a<c<d<b</math>";<ref name=J1 /> but often a weaker statement suffices, assuming only that <math>h</math> (or <math>h</math> and a number of its derivatives) vanishes at the endpoints <math>a</math>, <math>b</math>;<ref name=GF1 /> in this case the closed interval <math>[a,b]</math> is used.

==Version for two given functions==

:If a pair of continuous functions ''f'', ''g'' on an interval (''a'',''b'') satisfies the equality
::<math> \int_a^b ( f(x) \, h(x) + g(x) \, h'(x) ) \, \mathrm{d}x = 0 </math>
:for all compactly supported smooth functions ''h'' on (''a'',''b''), then ''g'' is differentiable, and ''g''' = ''f'' &nbsp;everywhere.<ref name=GF4>{{harvnb|Gelfand|Fomin|1963|loc=Lemma 4 on p.11}}</ref><ref name=H1>{{harvnb|Hestenes|1966|loc=Lemma 15.1 on p.50}}</ref>

The special case for ''g'' = 0 is just the basic version.

Here is the special case for ''f'' = 0 (often sufficient).

:If a continuous function ''g'' on an interval (''a'',''b'') satisfies the equality
::<math> \int_a^b g(x) \, h'(x) \, \mathrm{d}x = 0 </math>
:for all smooth functions ''h'' on (''a'',''b'') such that <math>h(a)=h(b)=0</math>, then ''g'' is [[Constant function|constant]].<ref name=GF2>{{harvnb|Gelfand|Fomin|1963|loc=Lemma 2 on p.10}}</ref>

If, in addition, continuous differentiability of ''g'' is assumed, then [[Integration by parts|integration by parts]] reduces both statements to the basic version; this case is attributed to [[Biography:Joseph-Louis Lagrange|Joseph-Louis Lagrange]], while the proof of differentiability of ''g'' is due to Paul du Bois-Reymond.

==Versions for discontinuous functions==

The given functions (''f'', ''g'') may be discontinuous, provided that they are locally integrable (on the given interval). In this case, [[Lebesgue integration]] is meant, the conclusions hold [[Almost everywhere|almost everywhere]] (thus, in all continuity points), and differentiability of ''g'' is interpreted as local [[Absolute continuity|absolute continuity]] (rather than continuous differentiability).<ref name=J2>{{harvnb|Jost|Li-Jost|1998|loc=Lemma 1.2.1 on p.13}}</ref><ref name=Gia>{{harvnb|Giaquinta|Hildebrandt|1996|loc=section 2.3: Mollifiers}}</ref> Sometimes the given functions are assumed to be piecewise continuous, in which case Riemann integration suffices, and the conclusions are stated everywhere except the finite set of discontinuity points.<ref name=H1 />

==Higher derivatives==

:If a tuple of continuous functions <math>f_0,f_1,\dots,f_n</math> on an interval (''a'',''b'') satisfies the equality
::<math> \int_a^b ( f_0(x) \, h(x) + f_1(x) \, h'(x) + \dots + f_n(x) \, h^{(n)}(x) ) \, \mathrm{d}x = 0 </math>
:for all compactly supported smooth functions ''h'' on (''a'',''b''), then there exist continuously differentiable functions <math> u_0,u_1,\dots,u_{n-1} </math> on (''a'',''b'') such that
::<math> 
\begin{align}
f_0 &= u'_0,  \\
f_1 &= u_0 + u'_1, \\
f_2 &= u_1 + u'_2 \\
\vdots \\
f_{n-1} &= u_{n-2} + u'_{n-1}, \\
f_n &= u_{n-1} 
\end{align}
</math>
:everywhere.<ref name=H2>{{harvnb|Hestenes|1966|loc=Lemma 13.1 on p.105}}</ref>

This necessary condition is also sufficient, since the integrand becomes <math> (u_0 h)' + (u_1 h')' + \dots + (u_{n-1} h^{(n-1)})'. </math>

The case ''n'' = 1 is just the version for two given functions, since <math> f=f_0=u'_0 </math> and <math> f_1=u_0, </math> thus, <math> f_0-f'_1 = 0. </math>

In contrast, the case ''n''=2 does not lead to the relation <math> f_0 - f'_1 + f''_2 = 0, </math> since the function <math> f_2 = u_1 </math> need not be differentiable twice. The sufficient condition <math> f_0 - f'_1 + f''_2 = 0 </math> is not necessary. Rather, the necessary and sufficient condition may be written as <math> f_0 - (f_1 - f'_2)' = 0 </math> for ''n''=2, <math> f_0 - (f_1 - (f_2-f'_3)')' = 0 </math> for ''n''=3, and so on; in general, the brackets cannot be opened because of non-differentiability.

==Vector-valued functions==

Generalization to vector-valued functions <math>(a,b)\to\mathbb{R}^d</math> is straightforward; one applies the results for scalar functions to each coordinate separately,<ref>{{harvnb|Gelfand|Fomin|1963|loc=p.35}}</ref> or treats the vector-valued case from the beginning.<ref>{{harvnb|Jost|Li-Jost|1998}}</ref>

==Multivariable functions==

:If a continuous [[Function of several real variables|multivariable function]] ''f'' on an open set <math>\Omega\subset\mathbb{R}^d</math> satisfies the equality
::<math> \int_\Omega f(x) \, h(x) \, \mathrm{d}x = 0 </math>
:for all compactly supported smooth functions ''h'' on Ω, then ''f'' is identically zero.

Similarly to the basic version, one may consider a continuous function ''f'' on the closure of Ω, assuming that ''h'' vanishes on the boundary of Ω (rather than compactly supported).<ref>{{harvnb|Gelfand|Fomin|1963|loc=Lemma on p.22}}; the proof applies in both situations.</ref>

Here is a version for discontinuous multivariable functions.

:Let <math>\Omega\subset\mathbb{R}^d</math> be an open set, and <math>f\in L^2(\Omega)</math> satisfy the equality
::<math> \int_\Omega f(x) \, h(x) \, \mathrm{d}x = 0 </math>
:for all compactly supported smooth functions ''h'' on Ω. Then ''f''=0 (in ''L''<sup>2</sup>, that is, almost everywhere).<ref>{{harvnb|Jost|Li-Jost|1998|loc=Lemma 3.2.3 on p.170}}</ref>

==Applications==
This lemma is used to prove that [[Maxima and minima|extrema]] of the [[Functional (mathematics)|functional]]
:<math> J[y] = \int_{x_0}^{x_1} L(t,y(t),\dot y(t)) \, \mathrm{d}t </math>
are  [[Weak solution|weak solution]]s <math>y:[x_0,x_1]\to V</math> (for an appropriate vector space <math>V</math>) of the [[Euler–Lagrange equation]]
:<math> {\partial L(t,y(t),\dot y(t)) \over \partial y} = {\mathrm{d} \over \mathrm{d}t} {\partial L(t,y(t),\dot y(t)) \over \partial \dot y} .</math>
The Euler–Lagrange equation plays a prominent role in [[Physics:Classical mechanics|classical mechanics]] and [[Differential geometry|differential geometry]].

==Notes==
<references />

==References==
*{{citation|last1=Jost|first1=Jürgen|last2=Li-Jost|first2=Xianqing|title=Calculus of variations|publisher=Cambridge University|year=1998}}
*{{citation|last1=Gelfand|first1=I.M.|last2=Fomin|first2=S.V.|title=Calculus of variations|publisher=Prentice-Hall|year=1963}} (transl. from Russian).
*{{citation|last=Hestenes|first=Magnus R.|title=Calculus of variations and optimal control theory|publisher=John Wiley|year=1966}}
*{{citation|last1=Giaquinta|first1=Mariano|last2=Hildebrandt|first2=Stefan|title=Calculus of Variations I|publisher=Springer|year=1996}}



[[Category:Classical mechanics]]
[[Category:Calculus of variations]]
[[Category:Smooth functions]]

{{Sourceattribution|Fundamental lemma of calculus of variations|1}}