Order reduction, projectability and constraints of second-order field theories and higher-order mechanics

TL;DR

This paper investigates the geometric structure and constraints of second- and higher-order field theories, focusing on order reduction, projectability of forms, and the implications for Einstein's equations in vacuum.

Contribution

It provides a detailed analysis of the projectability and constraints in higher-order Lagrangian theories, applying the results to Einstein's equations and higher-order mechanics.

Findings

Poincaré-Cartan forms are projectable due to Lagrangian degeneracy.

Constraints derived from the Euler-Lagrange equations explain the order reduction.

Einstein vacuum equations emerge from the constraint algorithm applied to the Hilbert Lagrangian.

Abstract

The projectability of Poincar\'e-Cartan forms in a third-order jet bundle $J^3\pi$ onto a lower-order jet bundle is a consequence of the degenerate character of the corresponding Lagrangian. This fact is analyzed using the constraint algorithm for the associated Euler-Lagrange equations in $J^3\pi$. The results are applied to study the Hilbert Lagrangian for the Einstein equations (in vacuum) from a multisymplectic point of view. Thus we show how these equations are a consequence of the application of the constraint algorithm to the geometric field equations, meanwhile the other constraints are related with the fact that this second-order theory is equivalent to a first-order theory. Furthermore, the case of higher-order mechanics is also studied as a particular situation.