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  "body": "\n## Summary\n\nThe Euler–Poisson equation is the [[Euler–Lagrange equation|PS0VV149ET]] of an integral functional whose Lagrangian depends on derivatives of order higher than one.\nIf the Lagrangian involves derivatives up to order $k$, the equation is an ordinary differential equation of order at most $2k$.\nThe order is exactly $2k$ when the Lagrangian really depends on the $k$-th derivative.\nIt governs, for example, the bending of beams and the shape of cubic splines.\n\n## Explanations\n\n### L1 human\n\nSome energies depend on how much a curve bends, not only on its slope.\nBending is measured by the second derivative, so the energy contains $y''$.\nAsking that small changes of the curve do not lower the energy gives an equation of order four instead of two.\nFor the energy $\\frac{1}{2} \\int y''(x)^2 \\, dx$ the equation is $y'''' = 0$, whose solutions are the cubic polynomials.\n",
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