Euler–Poisson equation for higher derivatives

Summary

The Euler–Poisson equation is the Euler–Lagrange equation of an integral functional whose Lagrangian depends on derivatives of order higher than one. If the Lagrangian involves derivatives up to order $k$, the equation is an ordinary differential equation of order at most $2k$. The order is exactly $2k$ when the Lagrangian really depends on the $k$-th derivative. It governs, for example, the bending of beams and the shape of cubic splines.