Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems

Abstract

Pluri-Lagrangian systems are variational systems with the multi-dimensional consistency property. This notion has its roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics, in the theory of variational symmetries going back to Noether and in the theory of discrete integrable systems. A $d$-dimensional pluri-Lagrangian problem can be described as follows: given a $d$-form $L$ on an $m$-dimensional space, $m>d$, whose coefficients depend on a function $u$ of $m$ independent variables (called field), find those fields $u$ which deliver critical points to the action functionals $S_{\Sigma }={\int }_{\Sigma }L$ for any $d$-dimensional manifold $\Sigma$ in the $m$-dimensional space. We investigate discrete 2-dimensional linear pluri-Lagrangian systems, i.e. those with quadratic Lagrangians $L$. The action is a discrete analogue of the Dirichlet energy, and solutions are called discrete pluriharmonic functions. We classify linear pluri-Lagrangian systems with Lagrangians depending on diagonals. They are described by generalizations of the star-triangle map. Examples of more general quadratic Lagrangians are also considered.

arXiv commentarXiv admin note: text overlap with arXiv:1307.0523

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• On the Lagrangian structure of integrable hierarchies
• Variational symmetries and pluri-Lagrangian systems in classical mechanics
• Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems
• Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs
• Continuum limits of pluri-Lagrangian systems
• Discrete time Toda systems
• Lagrangian 3-form structure for the Darboux system and the KP hierarchy
• Lagrangian multiforms on Lie groups and non-commuting flows
• Hamiltonian structures for integrable hierarchies of Lagrangian PDEs