Lagrangian Theory of Constrained Systems: Cosmological Application

TL;DR

This paper develops a covariant Lagrangian framework for analyzing constrained gravitational systems, revealing differences between first- and second-order formalisms in cosmological models with torsion.

Contribution

It introduces a generalized Lagrangian method applicable to implicit differential equations, uncovering second-generation constraints unique to second-order formalism in cosmology.

Findings

Second-order formalism yields additional constraints not seen in first-order.

The Lagrangian approach provides a covariant analysis of constrained gravity models.

Differences between formalisms impact physical predictions in cosmological scenarios.

Abstract

Previous work in the literature has studied the Hamiltonian structure of an R-squared model of gravity with torsion in a closed Friedmann-Robertson-Walker universe. Within the framework of Dirac's theory, torsion is found to lead to a second-class primary constraint linear in the momenta and a second-class secondary constraint quadratic in the momenta. This paper studies in detail the same problem at a Lagrangian level, i.e. working on the tangent bundle rather than on phase space. The corresponding analysis is motivated by a more general program, aiming to obtain a manifestly covariant, multisymplectic framework for the analysis of relativistic theories of gravitation regarded as constrained systems. After an application of the Gotay-Nester Lagrangian analysis, the paper deals with the generalized method, which has the advantage of being applicable to any system of differential equations in implicit form. Multiplication of the second-order Lagrange equations by a vector with zero eigenvalue for the Hessian matrix yields the so-called first-generation constraints. Remarkably, in the cosmological model here considered, if Lagrange equations are studied using second-order formalism, a second-generation constraint is found which is absent in first-order formalism. This happens since first- and second-order formalisms are inequivalent. There are, however, no {\it a priori} reasons for arguing that one of the two is incorrect. First- and second-generation constraints are used to derive physical predictions for the cosmological model.