Summary
Previous work in the literature has studied the Hamiltonian structure of anR 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 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, noa 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.
Similar content being viewed by others
References
G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo andC. Rubano:Phys. Rep.,188, 147 (1990).
G. Esposito:Nuovo Cimento B,104, 199 (1989).
E. Sezgin andP. van Nieuwenhuizen:Phys. Rev. D.,21, 3269 (1980).
I. A. Nikolic:Phys. Rev. D,30, 2508 (1984).
V. Szczyrba:Phys. Rev. D,36, 351 (1987).
V. Szczyrba:J. Math. Phys. (N.Y.),28, 146 (1987).
G. T. Horowitz:Phys. Rev. D.,31, 1169 (1985).
G. Esposito:Nuovo Cimento B,106, 1315 (1991).
M. A. H. Maccallum:Anisotropic and inhomogeneous relativistic cosmologies, inGeneral Relativity, an Einstein Centenary Survey, edited byS. W. Hawking andW. Israel (Cambridge University Press, Cambridge, 1979).
D. Boulware:Quantization of higher-derivative theories of gravity, inQuantum Theory of Gravity, edited byS. M. Christensen (Adam Hilger, Bristol, 1984).
P. A. M. Dirac:Lectures on Quantum Mechanics (Belfer Graduate School of Science, Yeshiva University, New York, N.Y., 1964).
G. Marmo, N. Mukunda andJ. Samuel:Riv. Nuovo Cimento,6, 1 (1983).
G. Esposito:Quantum Gravity Quantum Cosmology and Lorentzian Geometries, second corrected and enlarged edition,Lect. Notes Phys., New Ser. m: Monographs, Vol. m12 (Springer-Verlag, Berlin, 1994).
M. J. Gotay andJ. M. Nester:Ann. Inst. H. Poincaré A,30, 129 (1979).
J. F. Carinena, C. Lopez andN. Roman-Roy:J. Geom. Phys.,4, 315 (1987).
G. Marmo, E. J. Saletan, A. Simoni andB. Vitale:Dynamical Systems, a Differential Geometric Approach to Symmetry and Reduction (John Wiley, New York, N.Y., 1985).
G. Mendella:Geometrical aspects of Lagrangian dynamical systems with Dirac constraints, Thesis, University of Napoli (1987);A Geometrical Formalism for Constraints’ Theory, PhD Thesis, University of Napoli (1992).
G. Marmo, G. Mendella andW. M. Tulczyjew:Integrability of implicit differential equations, to be published inJ. Phys. A.
K. Sundermeyer:Constrained Dynamics (Springer-Verlag, Berlin, 1982).
E. Kamke:Differentialgleichungen Lösungsmethoden und Lösungen (Chelsea Publishing Company, New York, N.Y., 1959).
M. E. V. Costa, H. D. Girotti andT. J. M. Simoes:Phys. Rev. D.,32, 405 (1985).
M. Gotay, J. Isenberg, J. Marsden, R. Montgomery, J. Sniatycki andP. B. Yasskin:Momentum maps and the Hamiltonian treatment of classical field theories with constraints to appear inMathematical Sciences Research Institute Publications (Springer-Verlag, Berlin).
A. Ashtekar:Lectures on Non-Perturbative Canonical Gravity (World Scientific, Singapore, 1991).
M. Ferraris, M. Francaviglia andI. Sinicco:Nuovo Cimento B,107, 1303 (1992).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Esposito, G., Gionti, G., Marmo, G. et al. Lagrangian theory of constrained systems: Cosmological application. Nuov Cim B 109, 1259–1273 (1994). https://doi.org/10.1007/BF02722837
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02722837