Euler–Lagrange equations and boundary terms

Assume the integrand is twice continuously differentiable near the candidate graph, the candidate is twice continuously differentiable, and perturbations are continuously differentiable with fixed endpoint values. These classical hypotheses justify the chain rule and integration by parts on the compact interval. Start with compactly supported smooth perturbations, which vanish near both endpoints. If stationarity holds for all of them, the continuous coefficient multiplying the perturbation after integration by parts must vanish in the interior.

The fundamental lemma explains this last step. If a continuous residual were positive at some interior point, it would stay positive on a small interval. A nonnegative smooth perturbation supported there would make its integral against the residual positive, contradicting stationarity. Negative residuals are treated by changing the sign. Thus an integral identity for every localized perturbation becomes a pointwise differential equation. With less regular residuals, the corresponding statement is almost-everywhere or distributional, not automatically classical.

When computing $F_u$ and $F_v$, treat the position and slope arguments as independent variables. Only afterward substitute the candidate function and its derivative. The derivative with respect to the interval coordinate is then a total derivative along that candidate, so it can involve the second derivative of the function. Mixing these two operations can omit terms or incorrectly differentiate a coefficient twice. Writing the integrand with a separate slope symbol is a simple way to keep the operations distinct.

For $F=\tfrac12v^2-f(x)u$, the equation is $-u''=f$. The negative sign reflects integration by parts, and can be checked by testing a positive load between zero endpoints: the minimizing displacement is positive and concave downward. If the function is only in $H^1$, retain the weak form $\int u'h'=\int fh$ for all homogeneous test functions. This requires no classical second derivative and is the natural starting point for finite-element approximations and existence arguments.

An Euler–Lagrange solution need not minimize a nonconvex functional. For the derivative-energy functional with a linear load, however, an exact expansion around a stationary candidate proves a stronger conclusion. Every competitor is the candidate plus a homogeneous perturbation. The linear term vanishes by weak stationarity, leaving one half the squared derivative norm. This quantity is nonnegative and is zero only for a constant perturbation, which the zero endpoint traces force to be zero. The stationary candidate is therefore the unique global minimizer.

Notice the order of the reasoning: solve or identify a candidate, verify the weak equation, then compare the full objective for arbitrary competitors. If instead you only check one convenient family of trial functions, your result proves optimality inside that family. General existence can be established even without an explicit formula, but requires compactness and lower semicontinuity rather than a claim that a differential equation ought to have a solution. These distinctions become important for nonlinear energies, rough coefficients, and higher-dimensional problems.