Vấn đề bậc hai đối với lý thuyết trường Lagrangian k-presymplectic: ứng dụng cho mô hình Einstein–Palatini
Tóm tắt
Từ khóa
Tài liệu tham khảo
Abraham, R., Marsden, J.E.: Foundations of Mechanics. Advanced Book Program, 2nd edn. Benjamin/Cummings Publishing Co., Inc., Reading (1978). https://doi.org/10.1090/chel/364
Anderson, J.L., Bergmann, P.G.: Constraints in covariant field theories. Phys. Rev. 83, 1018–1025 (1951). https://doi.org/10.1103/PhysRev.83.1018
Arnol’d, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60, 2nd edn. Springer, New York (1989). https://doi.org/10.1007/978-1-4757-2063-1
Awane, A.: $$k$$-symplectic structures. J. Math. Phys. 33(12), 4046–4052 (1992). https://doi.org/10.1063/1.529855
Batlle, C., Gomis, J., Pons, J.M., Román-Roy, N.: Equivalence between the Lagrangian and Hamiltonian formalism for constrained systems. J. Math. Phys. 27(12), 2953–2962 (1986). https://doi.org/10.1063/1.527274
Batlle, C., Gomis, J., Pons, J.M., Román-Roy, N.: Lagrangian and Hamiltonian constraints. Lett. Math. Phys. 13(1), 17–23 (1987). https://doi.org/10.1007/BF00570763
Capriotti, S.: Differential geometry, Palatini gravity and reduction. J. Math. Phys. 55(1), 012902 (2014). https://doi.org/10.1063/1.4862855
Capriotti, S.: Unified formalism for Palatini gravity. Int. J. Geom. Methods Mod. Phys. 15(3), 1850044 (2018). https://doi.org/10.1142/S0219887818500445
Cariñena, J.F., López, C., Román-Roy, N.: Geometric study of the connection between the Lagrangian and Hamiltonian constraints. J. Geom. Phys. 4(3), 315–334 (1987). https://doi.org/10.1016/0393-0440(87)90017-9
Cariñena, J.F., López, C., Román-Roy, N.: Origin of the Lagrangian constraints and their relation with the Hamiltonian formulation. J. Math. Phys. 29(5), 1143–1149 (1987). https://doi.org/10.1063/1.527955
Chinea, D., de León, M., Marrero, J.C.: The constraint algorithm for time-dependent Lagrangians. J. Math. Phys. 35(7), 3410–3447 (1994). https://doi.org/10.1063/1.530476
Crampin, M.: Tangent bundle geometry for Lagrangian dynamics. J. Phys. A Math. Gen. 16(16), 3755–3772 (1983). https://doi.org/10.1088/0305-4470/16/16/014
de León, M., Marín-Solano, J., Marrero, J.C.: A geometrical approach to classical field theories: a constraint algorithm for singular theories. In: New Developments in Differential Geometry, vol. 350, pp. 291–312. Springer, Netherlands (1996). https://doi.org/10.1007/978-94-009-0149-0_22
de León, M., Marín-Solano, J., Marrero, J.C., Muñoz-Lecanda, M.C., Román-Roy, N.: Pre-multisymplectic constraint algorithm for field theories. Int. J. Geom. Methods Mod. Phys. 2(5), 839–871 (2005). https://doi.org/10.1142/S0219887805000880
de León, M., Marín-Solano, J., Marrero, J.C., Muñoz-Lecanda, M.C., Román-Roy, N.: Singular Lagrangian systems on jet bundles. Fortschr. Phys. 50(2), 105–169 (2002). https://doi.org/10.1002/1521-3978(200203)50:2<3c105::AID-PROP105>3e3.0.CO;2-N
de León, M., Méndez, I., Salgado, M.: Integrable $$p$$-almost tangent structures and tangent bundles of $$p^1$$-velocities. Acta Math. Hung. 58(1–2), 45–54 (1991). https://doi.org/10.1007/BF01903546
de León, M., Salgado, M., Vilariño, S.: Methods of Differential Geometry in Classical Field Theories: $$k$$-Symplectic and $$k$$-Cosymplectic Approaches. World Scientific, Hackensack (2016). https://doi.org/10.1142/9693
Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950). https://doi.org/10.4153/CJM-1950-012-1
Einstein, A.: Einheitliche Fieldtheorie von Gravitation und Elektrizität. Pruess. Akad.Wiss. 414, (1925); A. Unzicker and T. Case, “Translation of Einstein’s attempt of a unified field theory with teleparallelism”. arXiv:physics/0503046
Ferraris, M., Francaviglia, M., Reina, C.: Variational formulation of general relativity from 1915 to 1925 ‘Palatini’s Method’ discovered by Einstein in 1925. Gen. Relativ. Gravit. 14(3), 243–254 (1982). https://doi.org/10.1007/BF00756060
Gaset, J., Román-Roy, N.: Order reduction, projectability and constraints of second-order field theories and higher-order mechanics. Rep. Math. Phys. 78(3), 327–337 (2016). https://doi.org/10.1016/S0034-4877(17)30012-5
Gaset, J., Román-Roy, N.: New multisymplectic approach to the Metric–Affine (Einstein–Palatini) action for gravity. J. Geom. Mech. 11(3), 361–396 (2019). https://doi.org/10.3934/jgm.2019019
Gotay, M.J., Nester, J.M.: Presymplectic Lagrangian systems I: the constraint algorithm and the equivalence theorem. Ann. Inst. Henri Poincaré 30(2), 129–142 (1979). http://www.numdam.org/item?id=AIHPA_1979__30_2_129_0
Gotay, M.J., Nester, J.M.: Presymplectic Lagrangian systems. II: the second-order equation problem. Ann. Inst. Henri Poincaré 32(1), 1–13 (1980). http://www.numdam.org/item/AIHPA_1980_32_1_1_0/
Gotay, M.J., Nester, J.M., Hinds, G.: Presymplectic manifolds and the Dirac–Bergmann theory of constraints. J. Math. Phys. 19(11), 2388–2399 (1978). https://doi.org/10.1063/1.523597
Gràcia, X., Martín, R.: Geometric aspects of time-dependent singular differential equations. Int. J. Geom. Methods Mod. Phys. 2(4), 597–618 (2005). https://doi.org/10.1142/S0219887805000697
Gràcia, X., Martín, R., Román-Roy, N.: Constraint algorithm for $$k$$-presymplectic Hamiltonian systems: application to singular field theories. Int. J. Geom. Methods Mod. Phys. 6(5), 851–872 (2009). https://doi.org/10.1142/S0219887809003795
Gràcia, X., Pons, J.M.: A generalized geometric framework for constrained systems. Differ. Geom. Appl. 2, 223–247 (1992). https://doi.org/10.1016/0926-2245(92)90012-C
Gràcia, X., Rivas, X., Román-Roy, N.: Constraint algorithm for singular field theories in the k-cosymplectic framework. J. Geom. Mech. 12(1), 1–23 (2020). https://doi.org/10.3934/jgm.2020002
Günther, C.: The polysymplectic Hamiltonian formalism in the field theory and the calculus of variations. I. The local case. J. Differ. Geom. 25, 23–53 (1987). https://doi.org/10.4310/jdg/1214440723
Libermann, P., Marle, C.M.: Symplectic Geometry and Analytical Mechanics. Mathematics and its Applications, vol. 35. D. Reidel Publishing Co., Dordrecht (1987). https://doi.org/10.1007/978-94-009-3807-6
Marmo, G., Mendella, G., Tulczyjew, W.M.: Constrained Hamiltonian systems as implicit differential equations. J. Phys. A 30, 277–293 (1997). https://doi.org/10.1088/0305-4470/30/1/020
Munteanu, F., Rey, A.M., Salgado, M.: The Günther’s formalism in classical field theory: momentum map and reduction. J. Math. Phys. 45(5), 1730–1751 (2004). https://doi.org/10.1063/1.1688433
Muñoz-Lecanda, M.C., Román-Roy, N.: Lagrangian theory for presymplectic systems. Ann. Inst. Henry Poincaré Phys. Theor. 57, 27–45 (1992). http://www.numdam.org/item?id=AIHPA_1992__57_1_27_0
Palatini, A.: Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton. Rend. Circ. Mat. Palermo 43, 203–212 (1919). https://doi.org/10.1007/BF03014670
Muñoz-Masqué, J., Rosado, M.E.: Diffeomorphism-invariant covariant Hamiltonians of a pseudo-Riemannian metric and a linear connection. Adv. Theor. Math. Phys. 16(3), 851–886 (2012). https://oa.upm.es/15643
Rey, A.M., Román-Roy, N., Salgado, M.: Günther’s formalism in classical field theory: Skinner–Rusk approach and the evolution operator. J. Math. Phys. 46, 052901 (2005). https://doi.org/10.1063/1.1876872
Román-Roy, N.: Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories. Symmetry Integr. Geom. Methods Appl. (SIGMA) 5, 100 (2009). https://doi.org/10.3842/SIGMA.2009.100
Sarlet, W., Cantrijn, F., Crampin, M.: A new look at second-order equations and Lagrangian mechanics. J. Phys. A Math. Gen. 17(10), 1999–2009 (1984). https://doi.org/10.1088/0305-4470/17/10/012