Higher-order Euler-Lagrange equation

For the functionalthe vanishing of its first variation under fixed endpoint data gives

The Euler-Lagrange equation on an open interval can be derived using compactly supported variations even when an endpoint is singular. A formal solution with singular endpoint data need not make the improper action finite. For example, $y=x+x^{-1}$ solves $(x^4{y}^{\prime \prime }{)}^{\prime \prime }=0$, but substituting into $x^4(y^{\prime \prime }{)}^2+4y^2{y}^{\prime }$ gives $4x^2+4-4x^{-4}$, whose integral down to zero diverges. Stationarity of a finite functional then requires an additional regularization or admissible-space specification.

A perturbation preserving both values and slopes has $\eta =\eta \prime =0$ at the endpoints. Apply the Poincaré inequality to $\eta$ and then to $\eta \prime$. The resulting inequality makes a quadratic second variation proportional to $\int ({\eta }^2-(\eta \prime \prime {)}^2)$ nonpositive. If a functional is quadratic, its exact expansion about a stationary function then proves a global maximum.

Forwith the boundary conditions $y(0)=y^{\prime }(0)=0$, $y^{\prime \prime }(L)=0$, and $-A{y}^{\prime \prime \prime }(L)=F$, the higher-order Euler-Lagrange equation isIts solution splits as

For the clamped-free beam under uniform load and endpoint force, the additional minimized internal energy isIts derivative is the endpoint displacement caused by the endpoint force: