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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

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,", 2nd 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,", 2nd 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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