Here is the special case for f = 0 (often sufficient).If a continuous function g on an interval ( a, b) satisfies the equalityįor all compactly supported smooth functions h on ( a, b), then g is constant. The special case for g = 0 is just the basic version. ∫ a b ( f ( x ) h ( x ) + g ( x ) h ′ ( x ) ) d x = 0įor all compactly supported smooth functions h on ( a, b), then g is differentiable, and g' = f everywhere. Version for two given functionsIf a pair of continuous functions f, g on an interval ( a, b) satisfies the equality "Compactly supported" means "vanishes outside ( c, d ) for some c, d such that a < c < d < b" 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 is used. 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. Contentsīasic versionIf a continuous function f on an open interval ( a, b ) satisfies the equalityįor all compactly supported smooth functions h on ( a, b ), then f is identically zero.
More powerful versions are used when needed. Basic versions are easy to formulate and prove. Several versions of the lemma are in use. The proof usually exploits the possibility to choose δf concentrated on an interval on which f keeps sign (positive or negative).
#Fundamental lemma of calculus of variations free
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. Accordingly, the necessary condition of extremum (functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf. In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point.
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