Fundamental lemma of calculus of variations

In mathematics, specifically in the calculus of variations, 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 equal zero) appears in a weak formulation (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), 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.
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 formula_7 for some formula_8, formula_9 such that <math>a<c<d"; but often a weaker statement suffices, assuming only that formula_4 (or formula_4 and a number of its derivatives) vanishes at the endpoints formula_12, formula_13; in this case the closed interval formula_14 is used.
Version for two given functions.
The special case for "g" = 0 is just the basic version.
Here is the special case for "f" = 0 (often sufficient).
If, in addition, continuous differentiability of "g" is assumed, then integration by parts reduces both statements to the basic version; this case is attributed to 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 (thus, in all continuity points), and differentiability of "g" is interpreted as local absolute continuity (rather than continuous differentiability). 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.
Higher derivatives.
This necessary condition is also sufficient, since the integrand becomes formula_22
The case "n" = 1 is just the version for two given functions, since formula_23 and formula_24 thus, formula_25
In contrast, the case "n"=2 does not lead to the relation formula_26 since the function formula_27 need not be differentiable twice. The sufficient condition formula_28 is not necessary. Rather, the necessary and sufficient condition may be written as formula_29 for "n"=2, formula_30 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 formula_31 is straightforward; one applies the results for scalar functions to each coordinate separately, or treats the vector-valued case from the beginning.
Multivariable functions.
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.
Applications.
This lemma is used to prove that extrema of the functional
are weak solutions formula_38 (for an appropriate vector space formula_39) of the Euler–Lagrange equation
The Euler–Lagrange equation plays a prominent role in classical mechanics and differential geometry.