Fundamental Lemma of Calculus of Variations

The fundamental lemma of calculus of variations states that if is continuous on , with , and

for every smooth function with compact support in , then on . If a real were nonzero at an interior point, it would have a fixed nonzero sign on a smaller open interval. A nonzero nonnegative bump function supported in that interval would make the integral nonzero. For complex , the same argument applies to its real part and imaginary part.

In the calculus of variations, this lemma turns the vanishing of the first variation for all interior perturbations into the Euler-Lagrange differential equation.