American Institute of Mathematical Sciences

Advanced
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, revised and enlarged, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.  Google Scholar

[2]

Cédric M. Campos, "Geometric Methods in Classical Field Theory and Continuous Media," Ph.D thesis, Universidad Autónoma de Madrid, 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), 475207, 24 pp.  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), 303-330. 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), 345-374.  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), 112901, 30 pp.  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), 8461-8484. 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), 7402-7444. 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), 85-105. 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., Inc., River Edge, NJ, 1997.  Google Scholar

[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.  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), 375-390.  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), 375-395. 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), 1984-1998. 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), 505201, 23 pp.  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), 385-436. 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, Springer-Verlag, Berlin-New York, 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), 151-165.  Google Scholar

[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.  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 "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.  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 "New Developments in Differential Geometry" (Debrecen, 1994), Math. Appl., 350, Kluwer Acad. Publ., Dordrecht, (1996), 291-312.  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), 21-47. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

M. Modugno, Jet involution and prolongations of connections, Časopis Pěst. Mat., 114 (1989), 356-365.  Google Scholar

[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.  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), 113-137.  Google Scholar

[33]

D. J. Saunders, "The Geometry of Jet Bundles," London Mathematical Society Lecture Note Series, 142, Cambridge University Press, Cambridge, 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), A15-A18.  Google Scholar

[35]

W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A675-A678.  Google Scholar

[36]

W. M. Tulczyjew, A symplectic framework of linear field theories, Ann. Mat. Pura Appl. (4), 130 (1982), 177-195. doi: 10.1007/BF01761494.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition, revised and enlarged, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.  Google Scholar

[2]

Cédric M. Campos, "Geometric Methods in Classical Field Theory and Continuous Media," Ph.D thesis, Universidad Autónoma de Madrid, 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), 475207, 24 pp.  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), 303-330. 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), 345-374.  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), 112901, 30 pp.  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), 8461-8484. 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), 7402-7444. 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), 85-105. 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., Inc., River Edge, NJ, 1997.  Google Scholar

[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.  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), 375-390.  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), 375-395. 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), 1984-1998. 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), 505201, 23 pp.  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), 385-436. 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, Springer-Verlag, Berlin-New York, 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), 151-165.  Google Scholar

[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.  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 "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.  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 "New Developments in Differential Geometry" (Debrecen, 1994), Math. Appl., 350, Kluwer Acad. Publ., Dordrecht, (1996), 291-312.  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), 21-47. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

M. Modugno, Jet involution and prolongations of connections, Časopis Pěst. Mat., 114 (1989), 356-365.  Google Scholar

[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.  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), 113-137.  Google Scholar

[33]

D. J. Saunders, "The Geometry of Jet Bundles," London Mathematical Society Lecture Note Series, 142, Cambridge University Press, Cambridge, 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), A15-A18.  Google Scholar

[35]

W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A675-A678.  Google Scholar

[36]

W. M. Tulczyjew, A symplectic framework of linear field theories, Ann. Mat. Pura Appl. (4), 130 (1982), 177-195. doi: 10.1007/BF01761494.  Google Scholar

[1]

Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511
[2]

Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449
[3]

Menita Carozza, Jan Kristensen, Antonia Passarelli di Napoli. On the validity of the Euler-Lagrange system. Communications on Pure & Applied Analysis, 2015, 14 (1) : 51-62. doi: 10.3934/cpaa.2015.14.51
[4]

Katarzyna Grabowska, Luca Vitagliano. Tulczyjew triples in higher derivative field theory. Journal of Geometric Mechanics, 2015, 7 (1) : 1-33. doi: 10.3934/jgm.2015.7.1
[5]

Katarzyna Grabowska, Janusz Grabowski. Tulczyjew triples: From statics to field theory. Journal of Geometric Mechanics, 2013, 5 (4) : 445-472. doi: 10.3934/jgm.2013.5.445
[6]

David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319
[7]

Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577
[8]

Eduardo Martínez. Classical field theory on Lie algebroids: Multisymplectic formalism. Journal of Geometric Mechanics, 2018, 10 (1) : 93-138. doi: 10.3934/jgm.2018004
[9]

Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223
[10]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159
[11]

Katarzyna Grabowska, Marcin Zając. The Tulczyjew triple in mechanics on a Lie group. Journal of Geometric Mechanics, 2016, 8 (4) : 413-435. doi: 10.3934/jgm.2016014
[12]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[13]

Yuan Xu, Xin Jin, Saiwei Wang, Yang Tang. Optimal synchronization control of multiple euler-lagrange systems via event-triggered reinforcement learning. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1495-1518. doi: 10.3934/dcdss.2020377
[14]

Thomas Y. Hou, Danping Yang, Hongyu Ran. Multiscale analysis in Lagrangian formulation for the 2-D incompressible Euler equation. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1153-1186. doi: 10.3934/dcds.2005.13.1153
[15]

Zhihong Xia. Homoclinic points and intersections of Lagrangian submanifold. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 243-253. doi: 10.3934/dcds.2000.6.243
[16]

Jędrzej Śniatycki, Oǧul Esen. De Donder form for second order gravity. Journal of Geometric Mechanics, 2020, 12 (1) : 85-106. doi: 10.3934/jgm.2020005
[17]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013
[18]

Katarzyna Grabowska. Lagrangian and Hamiltonian formalism in Field Theory: A simple model. Journal of Geometric Mechanics, 2010, 2 (4) : 375-395. doi: 10.3934/jgm.2010.2.375
[19]

Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421
[20]

Hiroshi Morishita, Eiji Yanagida, Shoji Yotsutani. Structure of positive radial solutions including singular solutions to Matukuma's equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 871-888. doi: 10.3934/cpaa.2005.4.871

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (228)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences