-
Previous Article
Homogeneity and projective equivalence of differential equation fields
- JGM Home
- This Issue
- Next Article
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 |
References:
[1] |
R. Abraham and J. E. Marsden, "Foundations of Mechanics,", 2nd edition, (1978).
|
[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).
|
[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. |
[5] |
J. F. Cariñena, M. Crampin and A. Ibort, On the multisymplectic formalism for first order field theories,, Differential Geom. Appl., 1 (1991), 345.
|
[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).
|
[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. |
[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. |
[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. |
[10] |
G. Giachetta, L. Mangiarotti and G. Sardanashvily, "New Lagrangian and Hamiltonian Methods in Field Theory,", World Scientific Publishing Co., (1997).
|
[11] |
M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations. I. Covariant Hamiltonian formalism,, in, (1991), 203.
|
[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.
|
[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. |
[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. |
[18] |
E. Guzmán and J. C. Marrero, Time-dependent mechanics and Lagrangian submanifolds of presymplectic and Poisson manifolds,, J. Phys. A, 43 (2010).
|
[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. |
[21] |
J. Kijowski and W. M. Tulczyjew, "A Symplectic Framework for Field Theories,", Lecture Notes in Physics, 107 (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.
|
[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. |
[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.
|
[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.
|
[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. |
[28] |
M. de León and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics,", North-Holland Mathematics Studies, 158 (1989).
|
[29] |
M. Modugno, Jet involution and prolongations of connections,, Časopis Pěst. Mat., 114 (1989), 356.
|
[30] |
N. Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories,, SIGMA Symmetry Integrability Geom. Methods Appl., 5 (2009).
|
[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.
|
[33] |
D. J. Saunders, "The Geometry of Jet Bundles," London Mathematical Society Lecture Note Series, 142,, Cambridge University Press, (1989).
|
[34] |
W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne,, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976).
|
[35] |
W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne,, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976).
|
[36] |
W. M. Tulczyjew, A symplectic framework of linear field theories,, Ann. Mat. Pura Appl. (4), 130 (1982), 177.
doi: 10.1007/BF01761494. |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, "Foundations of Mechanics,", 2nd edition, (1978).
|
[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).
|
[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. |
[5] |
J. F. Cariñena, M. Crampin and A. Ibort, On the multisymplectic formalism for first order field theories,, Differential Geom. Appl., 1 (1991), 345.
|
[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).
|
[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. |
[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. |
[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. |
[10] |
G. Giachetta, L. Mangiarotti and G. Sardanashvily, "New Lagrangian and Hamiltonian Methods in Field Theory,", World Scientific Publishing Co., (1997).
|
[11] |
M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations. I. Covariant Hamiltonian formalism,, in, (1991), 203.
|
[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.
|
[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. |
[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. |
[18] |
E. Guzmán and J. C. Marrero, Time-dependent mechanics and Lagrangian submanifolds of presymplectic and Poisson manifolds,, J. Phys. A, 43 (2010).
|
[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. |
[21] |
J. Kijowski and W. M. Tulczyjew, "A Symplectic Framework for Field Theories,", Lecture Notes in Physics, 107 (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.
|
[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. |
[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.
|
[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.
|
[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. |
[28] |
M. de León and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics,", North-Holland Mathematics Studies, 158 (1989).
|
[29] |
M. Modugno, Jet involution and prolongations of connections,, Časopis Pěst. Mat., 114 (1989), 356.
|
[30] |
N. Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories,, SIGMA Symmetry Integrability Geom. Methods Appl., 5 (2009).
|
[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.
|
[33] |
D. J. Saunders, "The Geometry of Jet Bundles," London Mathematical Society Lecture Note Series, 142,, Cambridge University Press, (1989).
|
[34] |
W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne,, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976).
|
[35] |
W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne,, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976).
|
[36] |
W. M. Tulczyjew, A symplectic framework of linear field theories,, Ann. Mat. Pura Appl. (4), 130 (1982), 177.
doi: 10.1007/BF01761494. |
[1] |
Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete & Continuous Dynamical Systems - A, 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 - A, 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 - A, 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 - A, 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 - A, 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 - A, 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] |
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 |
[18] |
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 |
[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 |
2019 Impact Factor: 0.649
Tools
Metrics
Other articles
by authors
[Back to Top]