Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds

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March 2012, 4(1): 1-26. doi: 10.3934/jgm.2012.4.1

Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds

Cédric M. Campos ^1, , Elisa Guzmán ^2, and Juan Carlos Marrero ^2,

1.	Dept. Matemática Fundamental, Universidad de La Laguna, ULL, Avda. Astrofísico Fco. Sánchez, 38206 La Laguna, Tenerife, Spain
2.	ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Dept. Matemática Fundamental, Universidad de La Laguna, ULL, Avda. Astrofísico Fco. Sánchez, 38206 La Laguna, Tenerife, Spain, Spain

Received December 2011 Revised March 2012 Published April 2012

Abstract
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A description of classical field theories of first order in terms of Lagrangian submanifolds of premultisymplectic manifolds is presented. For this purpose, a Tulczyjew's triple associated with a fibration is discussed. The triple is adapted to the extended Hamiltonian formalism. Using this triple, we prove that Euler-Lagrange and Hamilton-De Donder-Weyl equations are the local equations defining Lagrangian submanifolds of a premultisymplectic manifold.

Keywords: Tulczyjew's triple, Field theory, multisymplectic structure, Lagrangian submanifold, Hamilton-De Donder-Weyl equation., Euler-Lagrange equation.

Mathematics Subject Classification: Primary: 70S05; Secondary: 70H03, 70H05, 53D1.

Citation: Cédric M. Campos, Elisa Guzmán, Juan Carlos Marrero. Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (1) : 1-26. doi: 10.3934/jgm.2012.4.1

References:

[1]	R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2^nd edition, revised and enlarged, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.
[2]	Cédric M. Campos, "Geometric Methods in Classical Field Theory and Continuous Media," Ph.D thesis, Universidad Autónoma de Madrid, 2010.
[3]	C. M. Campos, M. de León, D. Martín de Diego and J. Vankerschaver, Unambiguous formalism for higher order Lagrangian field theories, J. Phys. A, 42 (2009), 475207, 24 pp.
[4]	F. Cantrijn, A. Ibort and M. de León, On the geometry of multisymplectic manifolds, J. Austral. Math. Soc. Ser. A, 66 (1999), 303-330. doi: 10.1017/S1446788700036636.
[5]	J. F. Cariñena, M. Crampin and A. Ibort, On the multisymplectic formalism for first order field theories, Differential Geom. Appl., 1 (1991), 345-374.
[6]	A. Echeverría-Enríquez, M. de León, M. C. Muñoz-Lecanda and N. Román-Roy, Extended Hamiltonian systems in multisymplectic field theories, J. Math. Phys., 48 (2007), 112901, 30 pp.
[7]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of Hamiltonian field theories: Equations and symmetries, J. Phys. A, 32 (1999), 8461-8484. doi: 10.1088/0305-4470/32/48/309.
[8]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys., 41 (2000), 7402-7444. doi: 10.1063/1.1308075.
[9]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, On the multimomentum bundles and the Legendre maps in field theories, Rep. Math. Phys., 45 (2000), 85-105. doi: 10.1016/S0034-4877(00)88873-4.
[10]	G. Giachetta, L. Mangiarotti and G. Sardanashvily, "New Lagrangian and Hamiltonian Methods in Field Theory," World Scientific Publishing Co., Inc., River Edge, NJ, 1997.
[11]	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.
[12]	M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations. II. Space $+$ time decomposition, Differential Geom. Appl., 1 (1991), 375-390.
[13]	M. J. Gotay, J. E. Marsden, J. A. Isenberg, R. Montgomery, J. Śniatycki and P. B. Yasskin, "Momentum Maps and Classical Fields. Part I: Covariant Field Theory," preprint, 2004,, \arXiv{physics/9801019}., ().
[14]	M. J. Gotay, J. E. Marsden, J. A. Isenberg, R. Montgomery, J. Śniatycki and P. B. Yasskin, "Momentum Maps and Classical Fields. Part II: Canonical Analysis of Field Theories," preprint, 2004, \arXiv{math-ph/0411032}., ().
[15]	K. Grabowska, Lagrangian and Hamiltonian formalism in field theory: A simple model, J. Geom. Mech., 2 (2010), 375-395. doi: 10.3934/jgm.2010.2.375.
[16]	K. Grabowska, The Tulczyjew triple for classical fields,, preprint, ().
[17]	K. Grabowska, J. Grabowski and P. Urbański, AV-differential geometry: Euler-Lagrange equations, J. Geom. Phys., 57 (2007), 1984-1998. doi: 10.1016/j.geomphys.2007.04.003.
[18]	E. Guzmán and J. C. Marrero, Time-dependent mechanics and Lagrangian submanifolds of presymplectic and Poisson manifolds, J. Phys. A, 43 (2010), 505201, 23 pp.
[19]	E. Guzmán, J. C. Marrero and J. Vankerschaver, Lagrangian submanifolds and classical field theories of first order on Lie algebroids,, work in progress., ().
[20]	D. Iglesias, J. C. Marrero, E. Padrón and D. Sosa, Lagrangian submanifolds and dynamics on Lie affgebroids, Rep. Math. Phys., 57 (2006), 385-436. doi: 10.1016/S0034-4877(06)80029-7.
[21]	J. Kijowski and W. M. Tulczyjew, "A Symplectic Framework for Field Theories," Lecture Notes in Physics, 107, Springer-Verlag, Berlin-New York, 1979.
[22]	I. Kolář and M. Modugno, Natural maps on the iterated jet prolongation of a fibred manifold, Ann. Mat. Pura Appl. (4), 158 (1991), 151-165.
[23]	M. de León and E. A. Lacomba, Lagrangian submanifolds and higher-order mechanical systems, J. Phys. A, 22 (1989), 3809-3820. doi: 10.1088/0305-4470/22/18/019.
[24]	M. de León, E. A. Lacomba and P. R. Rodrigues, Special presymplectic manifolds, Lagrangian submanifolds and the Lagrangian-Hamiltonian systems on jet bundles, in "Proceedings of the First 'Dr. Antonio A. R. Monteiro' Congress on Mathematics" (Spanish) (Bahía Blanca, 1991), Univ. Nac. del Sur, Bahía Blanca, (1991), 103-122.
[25]	M. de León, J. Marin-Solano and J. C. Marrero, A geometrical approach to classical field theories: A constraint algorithm for singular theories, in "New Developments in Differential Geometry" (Debrecen, 1994), Math. Appl., 350, Kluwer Acad. Publ., Dordrecht, (1996), 291-312.
[26]	M. de León, D. Martín de Diego and A. Santamaría-Merino, Tulczyjew triples and Lagrangian submanifolds in classical field theories, Appl. Diff. Geom. Mech., (2003), 21-47.
[27]	M. de León and J. C. Marrero, Constrained time-dependent Lagrangian systems and Lagrangian submanifolds, J. Math. Phys., 34 (1993), 622-644. doi: 10.1063/1.530264.
[28]	M. de León and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics," North-Holland Mathematics Studies, 158, North-Holland Publishing Co., Amsterdam, 1989.
[29]	M. Modugno, Jet involution and prolongations of connections, Časopis Pěst. Mat., 114 (1989), 356-365.
[30]	N. Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories, SIGMA Symmetry Integrability Geom. Methods Appl., 5 (2009), Paper 100, 25 pp.
[31]	N. Román-Roy, Á. M. Rey, M. Salgado and S. Vilariño, $k$-cosymplectic classical field theories: Tulczyjew, Skinner-Rusk and Lie-algebroid formulations,, preprint, ().
[32]	N. Román-Roy, Á. M. Rey, M. Salgado and S. Vilariño, On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories, J. Geom. Mech., 3 (2011), 113-137.
[33]	D. J. Saunders, "The Geometry of Jet Bundles," London Mathematical Society Lecture Note Series, 142, Cambridge University Press, Cambridge, 1989.
[34]	W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A15-A18.
[35]	W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A675-A678.
[36]	W. M. Tulczyjew, A symplectic framework of linear field theories, Ann. Mat. Pura Appl. (4), 130 (1982), 177-195. doi: 10.1007/BF01761494.

show all references

References:

[1]	R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2^nd edition, revised and enlarged, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.
[2]	Cédric M. Campos, "Geometric Methods in Classical Field Theory and Continuous Media," Ph.D thesis, Universidad Autónoma de Madrid, 2010.
[3]	C. M. Campos, M. de León, D. Martín de Diego and J. Vankerschaver, Unambiguous formalism for higher order Lagrangian field theories, J. Phys. A, 42 (2009), 475207, 24 pp.
[4]	F. Cantrijn, A. Ibort and M. de León, On the geometry of multisymplectic manifolds, J. Austral. Math. Soc. Ser. A, 66 (1999), 303-330. doi: 10.1017/S1446788700036636.
[5]	J. F. Cariñena, M. Crampin and A. Ibort, On the multisymplectic formalism for first order field theories, Differential Geom. Appl., 1 (1991), 345-374.
[6]	A. Echeverría-Enríquez, M. de León, M. C. Muñoz-Lecanda and N. Román-Roy, Extended Hamiltonian systems in multisymplectic field theories, J. Math. Phys., 48 (2007), 112901, 30 pp.
[7]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of Hamiltonian field theories: Equations and symmetries, J. Phys. A, 32 (1999), 8461-8484. doi: 10.1088/0305-4470/32/48/309.
[8]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys., 41 (2000), 7402-7444. doi: 10.1063/1.1308075.
[9]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, On the multimomentum bundles and the Legendre maps in field theories, Rep. Math. Phys., 45 (2000), 85-105. doi: 10.1016/S0034-4877(00)88873-4.
[10]	G. Giachetta, L. Mangiarotti and G. Sardanashvily, "New Lagrangian and Hamiltonian Methods in Field Theory," World Scientific Publishing Co., Inc., River Edge, NJ, 1997.
[11]	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.
[12]	M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations. II. Space $+$ time decomposition, Differential Geom. Appl., 1 (1991), 375-390.
[13]	M. J. Gotay, J. E. Marsden, J. A. Isenberg, R. Montgomery, J. Śniatycki and P. B. Yasskin, "Momentum Maps and Classical Fields. Part I: Covariant Field Theory," preprint, 2004,, \arXiv{physics/9801019}., ().
[14]	M. J. Gotay, J. E. Marsden, J. A. Isenberg, R. Montgomery, J. Śniatycki and P. B. Yasskin, "Momentum Maps and Classical Fields. Part II: Canonical Analysis of Field Theories," preprint, 2004, \arXiv{math-ph/0411032}., ().
[15]	K. Grabowska, Lagrangian and Hamiltonian formalism in field theory: A simple model, J. Geom. Mech., 2 (2010), 375-395. doi: 10.3934/jgm.2010.2.375.
[16]	K. Grabowska, The Tulczyjew triple for classical fields,, preprint, ().
[17]	K. Grabowska, J. Grabowski and P. Urbański, AV-differential geometry: Euler-Lagrange equations, J. Geom. Phys., 57 (2007), 1984-1998. doi: 10.1016/j.geomphys.2007.04.003.
[18]	E. Guzmán and J. C. Marrero, Time-dependent mechanics and Lagrangian submanifolds of presymplectic and Poisson manifolds, J. Phys. A, 43 (2010), 505201, 23 pp.
[19]	E. Guzmán, J. C. Marrero and J. Vankerschaver, Lagrangian submanifolds and classical field theories of first order on Lie algebroids,, work in progress., ().
[20]	D. Iglesias, J. C. Marrero, E. Padrón and D. Sosa, Lagrangian submanifolds and dynamics on Lie affgebroids, Rep. Math. Phys., 57 (2006), 385-436. doi: 10.1016/S0034-4877(06)80029-7.
[21]	J. Kijowski and W. M. Tulczyjew, "A Symplectic Framework for Field Theories," Lecture Notes in Physics, 107, Springer-Verlag, Berlin-New York, 1979.
[22]	I. Kolář and M. Modugno, Natural maps on the iterated jet prolongation of a fibred manifold, Ann. Mat. Pura Appl. (4), 158 (1991), 151-165.
[23]	M. de León and E. A. Lacomba, Lagrangian submanifolds and higher-order mechanical systems, J. Phys. A, 22 (1989), 3809-3820. doi: 10.1088/0305-4470/22/18/019.
[24]	M. de León, E. A. Lacomba and P. R. Rodrigues, Special presymplectic manifolds, Lagrangian submanifolds and the Lagrangian-Hamiltonian systems on jet bundles, in "Proceedings of the First 'Dr. Antonio A. R. Monteiro' Congress on Mathematics" (Spanish) (Bahía Blanca, 1991), Univ. Nac. del Sur, Bahía Blanca, (1991), 103-122.
[25]	M. de León, J. Marin-Solano and J. C. Marrero, A geometrical approach to classical field theories: A constraint algorithm for singular theories, in "New Developments in Differential Geometry" (Debrecen, 1994), Math. Appl., 350, Kluwer Acad. Publ., Dordrecht, (1996), 291-312.
[26]	M. de León, D. Martín de Diego and A. Santamaría-Merino, Tulczyjew triples and Lagrangian submanifolds in classical field theories, Appl. Diff. Geom. Mech., (2003), 21-47.
[27]	M. de León and J. C. Marrero, Constrained time-dependent Lagrangian systems and Lagrangian submanifolds, J. Math. Phys., 34 (1993), 622-644. doi: 10.1063/1.530264.
[28]	M. de León and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics," North-Holland Mathematics Studies, 158, North-Holland Publishing Co., Amsterdam, 1989.
[29]	M. Modugno, Jet involution and prolongations of connections, Časopis Pěst. Mat., 114 (1989), 356-365.
[30]	N. Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories, SIGMA Symmetry Integrability Geom. Methods Appl., 5 (2009), Paper 100, 25 pp.
[31]	N. Román-Roy, Á. M. Rey, M. Salgado and S. Vilariño, $k$-cosymplectic classical field theories: Tulczyjew, Skinner-Rusk and Lie-algebroid formulations,, preprint, ().
[32]	N. Román-Roy, Á. M. Rey, M. Salgado and S. Vilariño, On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories, J. Geom. Mech., 3 (2011), 113-137.
[33]	D. J. Saunders, "The Geometry of Jet Bundles," London Mathematical Society Lecture Note Series, 142, Cambridge University Press, Cambridge, 1989.
[34]	W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A15-A18.
[35]	W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A675-A678.
[36]	W. M. Tulczyjew, A symplectic framework of linear field theories, Ann. Mat. Pura Appl. (4), 130 (1982), 177-195. doi: 10.1007/BF01761494.

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