Multisymplectic Geometry and Noether Symmetries for the Field Theories and the Relativistic Mechanics

Authors: H. Loumi-Fergane, A. Belaidi

Abstract:

The problem of symmetries in field theory has been analyzed using geometric frameworks, such as the multisymplectic models by using in particular the multivector field formalism. In this paper, we expand the vector fields associated to infinitesimal symmetries which give rise to invariant quantities as Noether currents for classical field theories and relativistic mechanic using the multisymplectic geometry where the Poincaré-Cartan form has thus been greatly simplified using the Second Order Partial Differential Equation (SOPDE) for multi-vector fields verifying Euler equations. These symmetries have been classified naturally according to the construction of the fiber bundle used.  In this work, unlike other works using the analytical method, our geometric model has allowed us firstly to distinguish the angular moments of the gauge field obtained during different transformations while these moments are gathered in a single expression and are obtained during a rotation in the Minkowsky space. Secondly, no conditions are imposed on the Lagrangian of the mechanics with respect to its dependence in time and in qⁱ, the currents obtained naturally from the transformations are respectively the energy and the momentum of the system.

Keywords: Field theories, relativistic mechanics, Lagrangian formalism, multisymplectic geometry, symmetries, Noether theorem, conservation laws.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1132214

References:

[1] A. Awane, “k-symplectic structures”, J. Math. Phys. 33 (1992), 4046-4052.
[2] A. Awane, M. Goze, “Pfaffian systems, k-symplectic systems”. Kluwer Academic Publishers, Dordrecht (2000).
[3] F. Munteanu, A. M. Rey, M. Salgado, “The Günther’s formalism in classical field theory: momentum map and reduction”, J. Math. Phys. 45 (5) (2004) 1730–1751.
[4] C. Günther, “The polysymplectic Hamiltonian formalism in field theory and calculus of variations I: The local case” J. Differential Geom. 25 (1987) 23-53.
[5] M. de León, E. Merino, J.A. Oubĩna, P. Rodrigues, M. Salgado, “Hamiltonian systems on k-cosymplectic manifolds”. J. Math. Phys. 39 (2) (1998) 876–893.
[6] M. de León, E. Merino, M. Salgado, “k-cosymplectic manifolds and Lagrangian field theories”. J. Math. Phys. 42 (5) (2001) 2092–2104.
[7] J. Kijowski, “A finite-dimensional canonical formalism in the classical field theory”, Comm. Math. Phys. 30 (1973), 99-128.
[8] J. Kijowski, W. Szczyrba, “Multisymplectic manifolds and the geometrical construction of the Poisson brackets in the classical field theory”. Géométrie symplectique et physique mathématique (Colloq. International C.N.R.S., Aix-en-Provence, 1974) (1974) 347-349.
[9] J. Kijowski, W. M. Tulczyjew, A symplectic framework for field theories. Lecture Notes in Physics, 107. Springer-Verlag, New York, 1979.
[10] J. Sniatycki, “On the geometric structure of classical field theory in Lagrangian formulation”, Math. Proc. Cambridge Philos. Soc. 68 (1970) 475-484.
[11] P. L. García, A. Pérez-Rendón, “Symplectic approach to the theory of quantized fields. I”. Comm. Math. Phys. 13 (1969) 24-44.
[12] P. L. García, A. Pérez-Rendón, “Symplectic approach to the theory of quantized fields, II”. Arch. Ratio. Mech. Anal. 43 (1971), 101-124.
[13] H. Goldschmidt, S. Sternberg, “The Hamilton-Cartan formalism in the calculus of variations”. Ann. Inst. Fourier 23 (1973), 203-267.
[14] G. Martin, “Dynamical structures fork-vector fields”, Internat. J. Theoret. Phys. 27 (1988), 571-585.
[15] G. Martin, “A Darboux theorem for multi-symplectic manifolds”, Lett. Math. Phys.16 (1988), 133-138.
[16] M. J. Gotay, “An exterior differential systems approach to the Cartan form. In: Symplectic geometry and mathematical physics” (Aix-en-Provence, 1990). Progr. Math. 99, Birkh ̈auser Boston, MA, 1991, 160-188.
[17] M. J. Gotay, “A multisymplectic framework for classical field theory and the calculus of variations, I. Covariant Hamiltonian formalism”. In: Mechanics, analysis and geometry: 200 years after Lagrange. North-Holland Delta Ser. North-Holland, Amsterdam, (1991) 203-235.
[18] M. J. Gotay, “A multisymplectic framework for classical field theory and the calculus of variations, II. Space -time decomposition”. Differential Geom. App. 1(1991), 375-390.
[19] M. J. Gotay, J. Isenberg, and J. E. Marsden, “Momentum Maps and Classical Relativistic Fields, Part I: Covariant Field Theory”, www.arxiv.org: (2004) physics/9801019.
[20] F. Cantrijn, A. Ibort, M. de León, “On the geometry of multisymplectic manifolds”, J. Austral. Math. Soc. Ser. A 66 (1999), 303-330.
[21] F. Cantrijn, A. Ibort, M. de León, “Hamiltonian structures on multisymplectic manifolds”, Rend. Sem. Mat. Univ. Politec. Torino, 54 (1996), 225-236.
[22] H. Loumi-Fergane, A. Belaidi, “Multisymplectic geometry and k-cosymplectic structure for the field theories and the relativistic mechanics”. Int. J. Geom. Methods Mod. Phys. 10, 1350001 (2013) (22 pages).
[23] N. Román-Roy, “K-symplectic formulation for field theories. An introduction to symmetries.” (Seminar given in 2007).
[24] N. Román-Roy, M. Salgado, S. Vilariño, “Symmetries and conservation laws in the Günther k-symplectic formalism of field theory”, Rev. Math. Phys. 19 (10) (2007), 1117-1147.
[25] J. C. Marrero, N. Román-Roy, M. Salgado, S. Vilariño, “On a Kind Noether Symmetries and Conservation Laws in k-Cosymplectic Field Theory”, J. Math. Phys, 52 (2011), 22901.
[26] A. Echeverría-Enríquez, M. C. Muñoz-Lecanda, N. Román-Roy, “Multivector field formulation of Hamiltonian field theories: Equations and Symmetries”, J. Phys. A. 32 (48) (1999), 8461-8484.
[27] M. de León, D. M.de Diego, A.S. Merino, “Symmetries in classical field theories”, Int. J. Geom. Meth. Mod. Phys.1 (5) (2004), 651-710.
[28] H. Loumi and M. Tahiri, Reports On Mathematical Physics 33 (1993), 367-373.
[29] A. Echeverría-Enríquez, M.C. Muñoz-Lecanda, N. Román-Roy, “Geometry of Lagrangian first-order classical field theories”, Forts. Phys 44 (1996), 235-280.
[30] A. Echeverría-Enríquez, M.C. Muñoz-Lecanda, N. Román-Roy, “Multivector fields and connections. Setting Lagrangian equations in field theories”, J. Math. Phys. 39 (9), (1998) 4578-4603.
[31] N. Román-Roy, 2005 “Multisymplectic lagrangian and Hamiltonian formalisms of first-order classical field theories” (Preprint math-phys /0506022).
[32] G. Arytyunov, “Classical Field Theory” Preprint typeset in JHEP Style.
[33] H. Erbin, “Théories des champs classiques” version: 12 février 2011.
[34] D. Farquet, “Théorème de Noether en QFT” EPFL-Physique.
[35] J.M. Raimond, “Théorie Classique des Champs”, Septembre 9, 2014.
[36] M. de León, D.M. de Diego, “Conservation Laws and Symmetries in Economic Growth Models: a Geometrical Approach”, Extracta Mathematicae. 13 (3) (1998) 335-348.