The second-order problem for $k$-presymplectic Lagrangian field theories. Application to the Einstein--Palatini model

TL;DR

This paper develops a geometric constraint algorithm using $k$-symplectic geometry to address the second-order problem in singular Lagrangian field theories and applies it to the Einstein-Palatini model of General Relativity.

Contribution

It introduces a novel geometric constraint algorithm for second-order PDEs in classical field theories and applies it to the Einstein-Palatini model.

Findings

Successfully identifies solution submanifolds for Euler-Lagrange equations

Splits constraints based on their origin in the geometric framework

Provides insights into the structure of the Einstein-Palatini model

Abstract

In general, the system of $2$nd-order partial differential equations made of the Euler-Lagrange equations of classical field theories are not compatible for singular Lagrangians. This is the so-called second-order problem. The first aim of this work is to develop a fully geometric constraint algorithm which allows us to find a submanifold where the Euler-Lagrange equations have solution, and split the constraints into two kinds depending on their origin. We do so using $k$-symplectic geometry, which is the simplest intrinsic description of classical field theories. As a second aim, the Einstein-Palatini model of General Relativity is studied using this algorithm.