= Singular endpoints in higher-order variational problems
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 $(x^4y'')''=0$, but substituting into $x^4(y'')^2+4y^2y'$ 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.
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