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jets; connections; homogeneous formalism; Hamilton equations; energy; gravity
We propose a suitable formulation of the Hamiltonian formalism for Field Theory in terms of Hamiltonian connections and multisymplectic forms where a composite fibered bundle, involving a line bundle, plays the role of an extended configuration bundle. This new approach can be interpreted as a suitable generalization to Field Theory of the homogeneous formalism for Hamiltonian Mechanics. As an example of application, we obtain the expression of a formal energy for a parametrized version of the Hilbert–Einstein Lagrangian and we show that this quantity is conserved.
[1] Bureš J., Vanžura J.: Unified treatment of multisymplectic 3-forms in dimension 6. preprint arXiv:math.DG/0405101.
[2] Cabras A., Kolář I.: Connections on some functional bundles. Czechoslovak Math. J. 45 (120) (1995), 529–548. MR 1344519 | Zbl 0851.58007
[3] Crnković C., Witten E.: Covariant description of canonical formalism in geometrical theories. Three hundred years of gravitation, Cambridge Univ. Press, Cambridge (1987), 676–684. MR 0920461 | Zbl 0966.81533
[4] Dedecker P.: On the generalization of symplectic geometry to multiple integrals in the calculus of variations. Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975), Lecture Notes in Math. 570, Springer, Berlin, 1977, 395–456. MR 0458478
[5] De León M., Marrero J. C., Martin de Diego D.: A new geometric setting for classical field theories. Classical and quantum integrability (Warsaw, 2001); Banach Center Publ. 59, Polish Acad. Sci., Warsaw, (2003), 189–209. MR 2003724
[6] Echeverria-Enriquez A., De León M., Munoz-Lecanda M. C., Roman-Roy N.: Hamiltonian systems in multisymplectic field theories. preprint arXiv:math-ph/0506003; Zbl 1152.81420
[7] Francaviglia M., Palese M., Winterroth E.: A new geometric proposal for the Hamiltonian description of classical field theories. Proc. VIII Int. Conf. Differential Geom. Appl., O. Kowalski et al. eds., 2001 Silesian University in Opava, 415–424. MR 1978795 | Zbl 1109.70310
Francaviglia M., Palese M., Winterroth E.: A general geometric setting for the energy of the gravitational field. Inst. Phys. Conf. Ser. 176, I. Ciufolini et al. eds., Taylor & Francis 2005, 391-395.
[8] Giachetta G., Mangiarotti L., Sardanashvili G.: New Lagrangian and Hamiltonian Methods in Field Theory. World Scientific, Singapore, 1997. MR 2001723
[9] Goldschmidt H., Sternberg S.: The Hamilton–Cartan formalism in the calculus of variations. Ann. Inst. Fourier, Grenoble 23 (1) (1973), 203–267. MR 0341531 | Zbl 0243.49011
[10] Grabowska K., Kijowski J.: Canonical gravity and canonical energy. Proc. VIII Int. Conf. Differential Geometry and Appl. (Opava, Czech Republic August 27–31, 2001) O. Kowalski, D. Krupka and J. Slovak eds., 2001 Silesian University in Opava, 261–274. MR 1978783
[11] Hélein F., Kouneiher J.: Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage-Dedecker versus De Donder-Weyl. Adv. Theor. Math. Phys. 8 (3) (2004), 565–601. MR 2105190 | Zbl 1115.70017
[12] Hélein F., Kouneiher J.: The notion of observable in the covariant Hamiltonian formalism for the calculus of variations with several variables. Adv. Theor. Math. Phys. 8 (4) (2004), 735–777. MR 2141500 | Zbl 1113.70023
[13] Kanatchikov I. V.: Canonical structure of classical field theory in the polymomentum phase space. Rep. Math. Phys. 41 (1) (1998), 49–90. MR 1617894 | Zbl 0947.70020
[14] Kanatchikov I. V.: Precanonical quantum gravity: quantization without the space-time decomposition. Internat. J. Theoret. Phys. 40 (6) (2001), 1121–1149. MR 1834872 | Zbl 0984.83026
[15] Kijowski J.: A finite-dimensional canonical formalism in the classical field theory. Comm. Math. Phys. 30 (1973), 99–128. MR 0334772
[16] Kijowski J.: Multiphase spaces and gauge in the calculus of variations. Bull. Acad. Polon. Sciences, Math. Astr. Phys. XXII (12) (1974), 1219–1225. MR 0370653 | Zbl 0302.49024
[17] Kijowski J., Szczyrba W.: A canonical structure for classical field theories. Comm. Math. Phys. 46 (2) (1976), 183–206. MR 0406247 | Zbl 0348.49017
[18] Kijowski J., Tulczyjew W. M.: A symplectic framework for field theories. Lecture Notes in Phys. 107, Springer-Verlag, Berlin-New York, 1979. MR 0549772 | Zbl 0439.58002
[19] Kolář I.: A geometrical version of the higher order Hamilton formalism in fibred manifolds. J. Geom. Phys. 1 (2) (1984), 127–137. MR 0794983
[20] Krupková O.: Hamiltonian field theory. J. Geom. Phys. 43 (2002), 93–132. Zbl 1016.37033
[21] Mangiarotti L., Sardanashvily G.: Gauge Mechanics. World Scientific, Singapore, 1998. MR 1689375
[22] Mangiarotti L., Sardanashvily G.: Connections in classical and quantum field theory. World Scientific, Singapore, 2000. MR 1764255 | Zbl 1053.53022
[23] Panák M., Vanžura J.: 3-forms and almost complex structures on 6-dimensional manifolds. preprint arXiv:math.DG/0305312.
[24] Rey A. M., Roman-Roy N., Salgado M.: Gunther’s formalism $(k$-symplectic formalism$)$ in classical field theory: Skinner-Rusk approach and the evolution operator. J. Math. Phys. 46 (2005), 052901, 24pp. MR 2143001
[25] Rey A. M., Roman-Roy N., Salgado M.: k-cosymplectic formalism in classical field theory: the Skinner–Rusk approach. preprint arXiv:math-ph/0602038.
[26] Sardanashvily G.: Generalized Hamiltonian formalism for field theory. Constraint systems. World Scientific, Singapore, 1995. MR 1376141
[27] Sardanashvily G.: Hamiltonian time–dependent mechanics. J. Math. Phys. 39 (5) (1998), 2714–2729. MR 1621455 | Zbl 1031.70508
[28] Saunders D. J.: The geometry of jet bundles. Cambridge University Press, Cambridge, 1989. MR 0989588 | Zbl 0665.58002
[29] Winterroth E.: A $K$–theory for certain multisymplectic vector bundles. Proc. VIII Int. Conf. Differential Geometry and Appl., O. Kowalski et al. eds., 2001 Silesian University in Opava, 153–162. MR 1978772 | Zbl 1034.55001
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