We investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form

$$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$

By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

中文翻译：

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二阶可积拉格朗日方程和WDVV方程

我们研究与形式为2D二阶Lagrangian的Euler–Lagrange方程的可积性

$$ \ begin {aligned} \ int f（u_ {xx}，u_ {xy}，u_ {yy}）\ \ mathrm {d} x \ mathrm {d} y。\ end {aligned} $$

通过推导拉格朗日密度f的可积性条件，构造了可通过基本函数，雅可比theta函数和对数表示的可积拉格朗日的例子。建立了二阶可积拉格朗日方程与WDVV方程的链接。还讨论了对3D二阶可积Lagrangian的推广。