Lagrangian theory of constrained systems: Cosmological application

Abstract

Summary. -Previous work in the literature has studied the Hamiltonian structure of an R 2 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 1

Space-time covariant form of Ashtekar’s constraints

“…The first constraints one actually evaluates correspond to the secondary first-class constraints of the Hamiltonian formalism. At least at a classical level, the Lagrangian theory of constrained systems is by now a rich branch of modern mathematical physics [4,20,21], although the majority of general relativists are more familiar with the Hamiltonian framework.…”

“…At a Lagrangian level, however, there are no primary constraints [4,20], since one deals with the pull-back on a manifold corresponding to the primary-constraint submanifold of the Hamiltonian formalism. The problem remains to derive the constraints (if any)…”

“…In ref. [20], we were able to use the Gotay-Nester Lagrangian analysis in a finite-dimensional model to derive the Lagrangian counterpart of secondary second-class constraints. However, we do not yet know whether that technique can be extended to our formulation of general relativity.…”

“…The first constraints one actually evaluates correspond to the secondary first-class constraints of the Hamiltonian formalism. At least at a classical level, the Lagrangian theory of constrained systems is by now a rich branch of modern mathematical physics [4,20,21], although the majority of general relativists are more familiar with the Hamiltonian framework.(iv) In the ADM formalism [5,6], the invariance group is not the whole diffeomorphism group, but a subgroup given by the Cartesian product of diffeomorphisms on the real line with the diffeomorphism group on spacelike three-surfaces. By contrast, in the Lagrangian approach, the invariance group of the theory is the full diffeomorphism group of fourdimensional Lorentzian space-time.…”