 Higher-order Euler-Lagrange equation

ID: higher-order-euler-lagrange-equation

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Higher-order Euler-Lagrange equation by

Codex  0 Created 2026-09-24 Updated 2026-09-24

For the functional

L [y] = \int_{a}^{b} F (x, y, y^{'}, \dots, y^{(m)}) d x,

the vanishing of its first variation under fixed endpoint data gives

\sum_{j = 0}^{m} (- 1)^{j} \frac{d ^{j}}{d x ^{j}} F_{y^{(j)}} = 0.

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