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

Cédric M. Campos; Elisa Guzmán; Juan Carlos Marrero

doi:10.3934/jgm.2012.4.1

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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, (1978). Google Scholar
[2]	Cédric M. Campos, "Geometric Methods in Classical Field Theory and Continuous Media,", Ph.D thesis, (2010). Google Scholar
[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). Google Scholar
[4]	F. Cantrijn, A. Ibort and M. de León, On the geometry of multisymplectic manifolds,, J. Austral. Math. Soc. Ser. A, 66 (1999), 303. doi: 10.1017/S1446788700036636. Google Scholar
[5]	J. F. Cariñena, M. Crampin and A. Ibort, On the multisymplectic formalism for first order field theories,, Differential Geom. Appl., 1 (1991), 345. Google Scholar
[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). Google Scholar
[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. doi: 10.1088/0305-4470/32/48/309. Google Scholar
[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. doi: 10.1063/1.1308075. Google Scholar
[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. doi: 10.1016/S0034-4877(00)88873-4. Google Scholar
[10]	G. Giachetta, L. Mangiarotti and G. Sardanashvily, "New Lagrangian and Hamiltonian Methods in Field Theory,", World Scientific Publishing Co., (1997). Google Scholar
[11]	M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations. I. Covariant Hamiltonian formalism,, in, (1991), 203. Google Scholar
[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. Google Scholar
[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}., (). Google Scholar
[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}., (). Google Scholar
[15]	K. Grabowska, Lagrangian and Hamiltonian formalism in field theory: A simple model,, J. Geom. Mech., 2 (2010), 375. doi: 10.3934/jgm.2010.2.375. Google Scholar
[16]	K. Grabowska, The Tulczyjew triple for classical fields,, preprint, (). Google Scholar
[17]	K. Grabowska, J. Grabowski and P. Urbański, AV-differential geometry: Euler-Lagrange equations,, J. Geom. Phys., 57 (2007), 1984. doi: 10.1016/j.geomphys.2007.04.003. Google Scholar
[18]	E. Guzmán and J. C. Marrero, Time-dependent mechanics and Lagrangian submanifolds of presymplectic and Poisson manifolds,, J. Phys. A, 43 (2010). Google Scholar
[19]	E. Guzmán, J. C. Marrero and J. Vankerschaver, Lagrangian submanifolds and classical field theories of first order on Lie algebroids,, work in progress., (). Google Scholar
[20]	D. Iglesias, J. C. Marrero, E. Padrón and D. Sosa, Lagrangian submanifolds and dynamics on Lie affgebroids,, Rep. Math. Phys., 57 (2006), 385. doi: 10.1016/S0034-4877(06)80029-7. Google Scholar
[21]	J. Kijowski and W. M. Tulczyjew, "A Symplectic Framework for Field Theories,", Lecture Notes in Physics, 107 (1979). Google Scholar
[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. Google Scholar
[23]	M. de León and E. A. Lacomba, Lagrangian submanifolds and higher-order mechanical systems,, J. Phys. A, 22 (1989), 3809. doi: 10.1088/0305-4470/22/18/019. Google Scholar
[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, (1991), 103. Google Scholar
[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, 350 (1996), 291. Google Scholar
[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. Google Scholar
[27]	M. de León and J. C. Marrero, Constrained time-dependent Lagrangian systems and Lagrangian submanifolds,, J. Math. Phys., 34 (1993), 622. doi: 10.1063/1.530264. Google Scholar
[28]	M. de León and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics,", North-Holland Mathematics Studies, 158 (1989). Google Scholar
[29]	M. Modugno, Jet involution and prolongations of connections,, Časopis Pěst. Mat., 114 (1989), 356. Google Scholar
[30]	N. Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories,, SIGMA Symmetry Integrability Geom. Methods Appl., 5 (2009). Google Scholar
[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, (). Google Scholar
[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. Google Scholar
[33]	D. J. Saunders, "The Geometry of Jet Bundles," London Mathematical Society Lecture Note Series, 142,, Cambridge University Press, (1989). Google Scholar
[34]	W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne,, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976). Google Scholar
[35]	W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne,, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976). Google Scholar
[36]	W. M. Tulczyjew, A symplectic framework of linear field theories,, Ann. Mat. Pura Appl. (4), 130 (1982), 177. doi: 10.1007/BF01761494. Google Scholar

show all references

References:

[1]	R. Abraham and J. E. Marsden, "Foundations of Mechanics,", 2^nd edition, (1978). Google Scholar
[2]	Cédric M. Campos, "Geometric Methods in Classical Field Theory and Continuous Media,", Ph.D thesis, (2010). Google Scholar
[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). Google Scholar
[4]	F. Cantrijn, A. Ibort and M. de León, On the geometry of multisymplectic manifolds,, J. Austral. Math. Soc. Ser. A, 66 (1999), 303. doi: 10.1017/S1446788700036636. Google Scholar
[5]	J. F. Cariñena, M. Crampin and A. Ibort, On the multisymplectic formalism for first order field theories,, Differential Geom. Appl., 1 (1991), 345. Google Scholar
[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). Google Scholar
[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. doi: 10.1088/0305-4470/32/48/309. Google Scholar
[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. doi: 10.1063/1.1308075. Google Scholar
[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. doi: 10.1016/S0034-4877(00)88873-4. Google Scholar
[10]	G. Giachetta, L. Mangiarotti and G. Sardanashvily, "New Lagrangian and Hamiltonian Methods in Field Theory,", World Scientific Publishing Co., (1997). Google Scholar
[11]	M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations. I. Covariant Hamiltonian formalism,, in, (1991), 203. Google Scholar
[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. Google Scholar
[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}., (). Google Scholar
[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}., (). Google Scholar
[15]	K. Grabowska, Lagrangian and Hamiltonian formalism in field theory: A simple model,, J. Geom. Mech., 2 (2010), 375. doi: 10.3934/jgm.2010.2.375. Google Scholar
[16]	K. Grabowska, The Tulczyjew triple for classical fields,, preprint, (). Google Scholar
[17]	K. Grabowska, J. Grabowski and P. Urbański, AV-differential geometry: Euler-Lagrange equations,, J. Geom. Phys., 57 (2007), 1984. doi: 10.1016/j.geomphys.2007.04.003. Google Scholar
[18]	E. Guzmán and J. C. Marrero, Time-dependent mechanics and Lagrangian submanifolds of presymplectic and Poisson manifolds,, J. Phys. A, 43 (2010). Google Scholar
[19]	E. Guzmán, J. C. Marrero and J. Vankerschaver, Lagrangian submanifolds and classical field theories of first order on Lie algebroids,, work in progress., (). Google Scholar
[20]	D. Iglesias, J. C. Marrero, E. Padrón and D. Sosa, Lagrangian submanifolds and dynamics on Lie affgebroids,, Rep. Math. Phys., 57 (2006), 385. doi: 10.1016/S0034-4877(06)80029-7. Google Scholar
[21]	J. Kijowski and W. M. Tulczyjew, "A Symplectic Framework for Field Theories,", Lecture Notes in Physics, 107 (1979). Google Scholar
[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. Google Scholar
[23]	M. de León and E. A. Lacomba, Lagrangian submanifolds and higher-order mechanical systems,, J. Phys. A, 22 (1989), 3809. doi: 10.1088/0305-4470/22/18/019. Google Scholar
[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, (1991), 103. Google Scholar
[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, 350 (1996), 291. Google Scholar
[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. Google Scholar
[27]	M. de León and J. C. Marrero, Constrained time-dependent Lagrangian systems and Lagrangian submanifolds,, J. Math. Phys., 34 (1993), 622. doi: 10.1063/1.530264. Google Scholar
[28]	M. de León and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics,", North-Holland Mathematics Studies, 158 (1989). Google Scholar
[29]	M. Modugno, Jet involution and prolongations of connections,, Časopis Pěst. Mat., 114 (1989), 356. Google Scholar
[30]	N. Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories,, SIGMA Symmetry Integrability Geom. Methods Appl., 5 (2009). Google Scholar
[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, (). Google Scholar
[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. Google Scholar
[33]	D. J. Saunders, "The Geometry of Jet Bundles," London Mathematical Society Lecture Note Series, 142,, Cambridge University Press, (1989). Google Scholar
[34]	W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne,, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976). Google Scholar
[35]	W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne,, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976). Google Scholar
[36]	W. M. Tulczyjew, A symplectic framework of linear field theories,, Ann. Mat. Pura Appl. (4), 130 (1982), 177. doi: 10.1007/BF01761494. Google Scholar

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