mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, specifically in the

calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

, a variation of a function can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum (

functional derivative In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on w ...

equal zero) appears in a

weak formulation Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or con ...

(variational form) integrated with an arbitrary function . The fundamental lemma of the calculus of variations is typically used to transform this weak formulation into the strong formulation (

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...

), free of the integration with arbitrary function. The proof usually exploits the possibility to choose concentrated on an interval on which 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

f

on an open interval

(a,b)

satisfies the equality ::

\int_a^b f(x)h(x)\,\mathrmx = 0

:for all

compactly supported In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalles ...

smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...

h

(a,b)

, then

f

is identically zero. Here "smooth" may be interpreted as "infinitely differentiable", but often is interpreted as "twice continuously differentiable" or "continuously differentiable" or even just "continuous", since these weaker statements may be strong enough for a given task. "Compactly supported" means "vanishes outside

,d /math> for some c, d such that a "; but often a weaker statement suffices, assuming only that h (or h and a number of its derivatives) vanishes at the endpoints a, b; in this case the closed interval,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 ::

\int_a^b ( f(x) \, h(x) + g(x) \, h'(x) ) \, \mathrmx = 0

:for all compactly supported smooth functions ''h'' on (''a'',''b''), then ''g'' is differentiable, and ''g = ''f'' everywhere. 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 ::

\int_a^b g(x) \, h'(x) \, \mathrmx = 0

:for all smooth functions ''h'' on (''a'',''b'') such that

h(a)=h(b)=0

, then ''g'' is constant. If, in addition, continuous differentiability of ''g'' is assumed, then

integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivat ...

reduces both statements to the basic version; this case is attributed to

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaPaul du Bois-Reymond.

Versions for discontinuous functions

The given functions (''f'', ''g'') may be discontinuous, provided that they are

locally integrable In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions li ...

(on the given interval). In this case,

Lebesgue integration In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...

is meant, the conclusions hold

almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...

(thus, in all continuity points), and differentiability of ''g'' is interpreted as local

absolute continuity In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...

(rather than continuous differentiability). Sometimes the given functions are assumed to be

piecewise continuous In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...

, in which case

Riemann integration In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Gö ...

suffices, and the conclusions are stated everywhere except the finite set of discontinuity points.

Higher derivatives

:If a tuple of continuous functions

f_0,f_1,\dots,f_n

on an interval (''a'',''b'') satisfies the equality ::

\int_a^b ( f_0(x) \, h(x) + f_1(x) \, h'(x) + \dots + f_n(x) \, h^(x) ) \, \mathrmx = 0

:for all compactly supported smooth functions ''h'' on (''a'',''b''), then there exist continuously differentiable functions

u_0,u_1,\dots,u_

on (''a'',''b'') such that ::

\begin
f_0 &= u'_0,  \\
f_1 &= u_0 + u'_1, \\
f_2 &= u_1 + u'_2 \\
\vdots \\
f_ &= u_ + u'_, \\
f_n &= u_ 
\end

:everywhere. This necessary condition is also sufficient, since the integrand becomes

(u_0 h)' + (u_1 h')' + \dots + (u_ h^)'.

The case ''n'' = 1 is just the version for two given functions, since

f=f_0=u'_0

and

f_1=u_0,

thus,

f_0-f'_1 = 0.

In contrast, the case ''n''=2 does not lead to the relation

f_0 - f'_1 + f''_2 = 0,

since the function

f_2 = u_1

need not be differentiable twice. The sufficient condition

f_0 - f'_1 + f''_2 = 0

is not necessary. Rather, the necessary and sufficient condition may be written as

f_0 - (f_1 - f'_2)' = 0

for ''n''=2,

f_0 - (f_1 - (f_2-f'_3)')' = 0

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 A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...

(a,b)\to\mathbb^d

is straightforward; one applies the results for scalar functions to each coordinate separately, or treats the vector-valued case from the beginning.

Multivariable functions

:If a continuous multivariable function ''f'' on an open set

\Omega\subset\mathbb^d

satisfies the equality ::

\int_\Omega f(x) \, h(x) \, \mathrmx = 0

: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). Here is a version for discontinuous multivariable functions. :Let

\Omega\subset\mathbb^d

be an open set, and

f\in L^2(\Omega)

satisfy the equality ::

\int_\Omega f(x) \, h(x) \, \mathrmx = 0

:for all compactly supported smooth functions ''h'' on Ω. Then ''f''=0 (in ''L''², that is, almost everywhere).

Applications

This lemma is used to prove that extrema of the

functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...

= \int_^ L(t,y(t),\dot y(t)) \, \mathrmt

are

weak solution In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precise ...

y:_0,x_1 to V

(for an appropriate vector space

V

) of the

Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...

=   .

The Euler–Lagrange equation plays a prominent role in

classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...

and differential geometry.

Notes

References

* * (transl. from Russian). * *{{citation, last1=Giaquinta, first1=Mariano, last2=Hildebrandt, first2=Stefan, title=Calculus of Variations I, publisher=Springer, year=1996 Classical mechanics Calculus of variations Smooth functions Lemmas in analysis