The fundamental lemma of the calculus of variations states that if f is a function in C [a,b], and In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

for every function h ∈ C²[a,b] with h(a) = h(b) = 0, then f(x) is identically zero in the open interval (a,b). In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...

This lemma is used to prove that a solution of the Euler-Lagrange equation In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than an independent statement, in and of itself. ... In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ...

is a stationary (and possibly extremal) "point" of the functional Stationary points (red pluses) and inflection points (green circles). ... A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some ε > 0 such that f(x*) ≥ f(x) for all x with |x-x*| < ε. ... In mathematics, the term functional is applied to certain functions. ...

External links

• Chapter III, Section 8: Proof of theorem 1 by Johan Byström, Lars-Erik Persson, and Fredrik Strömberg.