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

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.
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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