Second-order integrable Lagrangians and WDVV equations

TL;DR

This paper explores the integrability conditions of second-order 2D Lagrangians, constructs explicit examples, and establishes a connection to WDVV equations, with extensions to 3D cases.

Contribution

It derives integrability conditions for second-order Lagrangians and links them to WDVV equations, providing explicit examples involving special functions.

Findings

Explicit integrable Lagrangians constructed using elementary functions, theta functions, and dilogarithms.

Established a connection between second-order integrable Lagrangians and WDVV equations.

Discussed potential generalizations to three-dimensional cases.

Abstract

We investigate integrability of Euler-Lagrange equations associated with 2D second-order Lagrangians of the form \begin{equation*} \int f(u_{xx},u_{xy},u_{yy})\ dxdy. \end{equation*} 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.