March  2011, 3(1): 113-137. doi: 10.3934/jgm.2011.3.113

On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories

1. 

Departamento de Matemática Aplicada IV. Universitat Politècnica de Catalunya-BarcelonaTech., Edificio C-3, Campus Norte UPC, C/ Jordi Girona 1. 08034 Barcelona, Spain

2. 

Departamento de Xeometría e Topoloxía. Facultade de Matemáticas,, Universidade de Santiago de Compostela., 15706-Santiago de Compostela, Spain, Spain

3. 

Departamento de Matemáticas, Facultade de Ciencias, Universidad de A Coruña. 15071-A Coruña, Spain

Received  October 2010 Revised  March 2011 Published  April 2011

The objective of this work is twofold: First, we analyze the relation between the $k$-cosymplectic and the $k$-symplectic Hamiltonian and Lagrangian formalisms in classical field theories. In particular, we prove the equivalence between $k$-symplectic field theories and the so-called autonomous $k$-cosymplectic field theories, extending in this way the description of the symplectic formalism of autonomous systems as a particular case of the cosymplectic formalism in non-autonomous mechanics. Furthermore, we clarify some aspects of the geometric character of the solutions to the Hamilton-de Donder-Weyl and the Euler-Lagrange equations in these formalisms. Second, we study the equivalence between $k$-cosymplectic and a particular kind of multisymplectic Hamiltonian and Lagrangian field theories (those where the configuration bundle of the theory is trivial).
Citation: Narciso Román-Roy, Ángel M. Rey, Modesto Salgado, Silvia Vilariño. On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories. Journal of Geometric Mechanics, 2011, 3 (1) : 113-137. doi: 10.3934/jgm.2011.3.113
References:
[1]

R. A. Abraham and J. E. Marsden, "Foundations of Mechanics,'' 2nd Edition, Benjamin-Cummings Publishing Company, New York, 1978.  Google Scholar

[2]

A. Awane, $k$-symplectic structures, J. Math. Phys., 33 (1992), 4046-4052. doi: 10.1063/1.529855.  Google Scholar

[3]

A. Awane, $G$-spaces $k$-symplectic homogènes, J. Geom. Phys., 13 (1994), 139-157. doi: 10.1016/0393-0440(94)90024-8.  Google Scholar

[4]

A. Awane and M. Goze, "Pfaffian Systems, $k$-Symplectic Systems,'' Kluwer Acad. Pub., Dordrecht 2000.  Google Scholar

[5]

J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order field theories, Diff. Geom. Appl., 1 (1991), 345-374.  Google Scholar

[6]

D. Chinea, M. de León and J. C. Marrero, Locally conformal cosymplectic manifolds and time-dependent Hamiltonian systems, Comment. Math. Univ. Carolin., 32 (1991), 383-387.  Google Scholar

[7]

J. Dieudonné, "Foundations of Modern Analysis,'' 2nd ed., Academic Press, New York, 1969.  Google Scholar

[8]

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

[9]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector fields and connections. Setting Lagrangian equations for field theories, J. Math. Phys., 39 (1998), 4578-4603.  Google Scholar

[10]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of Hamiltonian field theories: Equations and symmetries, J. Phys. A: Math. Gen., 32 (1999), 8461-8484.  Google Scholar

[11]

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.  Google Scholar

[12]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, "New Lagrangian and Hamiltonian Methods in Field Theory,'' World Scientific Pub. Co., Singapore, 1997.  Google Scholar

[13]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, Covariant Hamilton equations for field theory, J. Phys. A, 32 (1999), 6629-6642. doi: 10.1088/0305-4470/32/38/302.  Google Scholar

[14]

M. J. Gotay, J. Isenberg, J. E. Marsden and R. Montgomery, Momentum maps and classical relativistic fields I: Covariant Theory, arXiv:physics/9801019v2, (1999). Google Scholar

[15]

C. Günther, The polysymplectic Hamiltonian formalism in field theory and calculus of variations I: The local case, J. Differential Geom., 25 (1987), 23-53.  Google Scholar

[16]

F. Hélein and J. Kouneiher, Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage-Dedecker versus De Donder-Weyl, Adv. Theor. Math. Phys., 8 (2004), 565-601.  Google Scholar

[17]

I. V. Kanatchikov, Canonical structure of classical field theory in the polymomentum phase space, Rep. Math. Phys., 41 (1998), 49-90. doi: 10.1016/S0034-4877(98)80182-1.  Google Scholar

[18]

J. Kijowski, A finite-dimensional canonical formalism in the classical field theory, Comm. Math. Phys., 30 (1973), 99-128. doi: 10.1007/BF01645975.  Google Scholar

[19]

J. Kijowski and W. Szczyrba, Multisymplectic manifolds and the geometrical construction of the Poisson brackets in the classical field theory, Géométrie Symplectique et Physique Mathématique Coll. Int. C.N.R.S., 237 (1975), 347-378.  Google Scholar

[20]

J. Kijowski and W. M. Tulczyjew, "A Symplectic Framework for Field Theories," Lect. Notes Phys., 170, Springer-Verlag, Berlin, 1979.  Google Scholar

[21]

J. M. Lee, "Introduction to Smooth Manifolds,'' Springer, New York, 2003.  Google Scholar

[22]

M. de León, J. Marín-Solano and J. C. Marrero, A geometrical approach to classical field theories: A constraint algorithm for singular theories, Proc. on New Developments in Differential geometry, L. Tamassi-J. Szenthe eds., Kluwer Acad. Press, (1996), 291-312.  Google Scholar

[23]

M. de León, M. McLean, L. K. Norris, A. Rey-Roca and M. Salgado, Geometric structures in field theory, arXiv:math-ph/0208036v1 (2002). Google Scholar

[24]

M. de León, E. Merino, J. A. Oubiña, P. Rodrigues and M. Salgado, Hamiltonian systems on $k$-cosymplectic manifolds, J. Math. Phys., 39 (1998), 876-893.  Google Scholar

[25]

M. de León, E. Merino and M. Salgado, $k$-cosymplectic manifolds and Lagrangian field theories, J. Math. Phys., 42 (2001), 2092-2104.  Google Scholar

[26]

M. McLean and L. K. Norris, Covariant field theory on frame bundles of fibered manifolds, J. Math. Phys., 41 (2000), 6808-6823. doi: 10.1063/1.1288797.  Google Scholar

[27]

J. E. Marsden and S. Shkoller, Multisymplectic geometry, covariant Hamiltonians and water waves, Math. Proc. Camb. Phil. Soc., 125 (1999), 553-575. doi: 10.1017/S0305004198002953.  Google Scholar

[28]

F. Munteanu, A. M. Rey and M. Salgado, The Günther's formalism in classical field theory: Momentum map and reduction, J. Math. Phys., 45 (2004), 1730-1751. doi: 10.1063/1.1688433.  Google Scholar

[29]

M.C. Muñoz-Lecanda, M. Salgado and S. Vilariño, $k$-symplectic and $k$-cosymplectic Lagrangian field theories: Some interesting examples and applications, Int. J. Geom. Meth. Mod. Phys., 7 (2010), 669-692. doi: 10.1142/S0219887810004506.  Google Scholar

[30]

L. K. Norris, Generalized symplectic geometry on the frame bundle of a manifold, Proc. Symp. Pure Math. 54, Part 2. Amer. Math. Soc., Providence RI, (1993), 435-465.  Google Scholar

[31]

L. K. Norris, $n$-symplectic algebra of observables in covariant Lagrangian field theory, J. Math. Phys., 42 (2001), 4827-4845. doi: 10.1063/1.1396835.  Google Scholar

[32]

C. Paufler and H. Römer, Geometry of Hamiltonian $n$-vector fields in multisymplectic field theory, J. Geom. Phys., 44 (2002), 52-69.  Google Scholar

[33]

A. M. Rey, N. Román-Roy and M. Salgado, Günther's formalism in classical field theory: Skinner-Rusk approach and the evolution operator, J. Math. Phys., 46 (2005), 052901. doi: 10.1063/1.1876872.  Google Scholar

[34]

N. Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories, Symmetry Integrability Geom. Methods Appl. (SIGMA), 5 (2009) 100, 25 pp.  Google Scholar

[35]

N. Román-Roy, M. Salgado and S. Vilariño, On a kind of Noether symmetries and conservation laws in $k$-symplectic field theory, J. Math. Phys., 52 (2011), 022901; (20 pages). Google Scholar

[36]

G. Sardanashvily, "Generalized Hamiltonian Formalism for Field Theory. Constraint Systems,'' World Scientific, Singapore, 1995.  Google Scholar

[37]

D. J. Saunders, "The Geometry of Jet Bundles,'' London Math. Soc. Lect. Notes Ser. 142, Cambridge, Univ. Press, 1989.  Google Scholar

show all references

References:
[1]

R. A. Abraham and J. E. Marsden, "Foundations of Mechanics,'' 2nd Edition, Benjamin-Cummings Publishing Company, New York, 1978.  Google Scholar

[2]

A. Awane, $k$-symplectic structures, J. Math. Phys., 33 (1992), 4046-4052. doi: 10.1063/1.529855.  Google Scholar

[3]

A. Awane, $G$-spaces $k$-symplectic homogènes, J. Geom. Phys., 13 (1994), 139-157. doi: 10.1016/0393-0440(94)90024-8.  Google Scholar

[4]

A. Awane and M. Goze, "Pfaffian Systems, $k$-Symplectic Systems,'' Kluwer Acad. Pub., Dordrecht 2000.  Google Scholar

[5]

J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order field theories, Diff. Geom. Appl., 1 (1991), 345-374.  Google Scholar

[6]

D. Chinea, M. de León and J. C. Marrero, Locally conformal cosymplectic manifolds and time-dependent Hamiltonian systems, Comment. Math. Univ. Carolin., 32 (1991), 383-387.  Google Scholar

[7]

J. Dieudonné, "Foundations of Modern Analysis,'' 2nd ed., Academic Press, New York, 1969.  Google Scholar

[8]

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

[9]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector fields and connections. Setting Lagrangian equations for field theories, J. Math. Phys., 39 (1998), 4578-4603.  Google Scholar

[10]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of Hamiltonian field theories: Equations and symmetries, J. Phys. A: Math. Gen., 32 (1999), 8461-8484.  Google Scholar

[11]

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.  Google Scholar

[12]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, "New Lagrangian and Hamiltonian Methods in Field Theory,'' World Scientific Pub. Co., Singapore, 1997.  Google Scholar

[13]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, Covariant Hamilton equations for field theory, J. Phys. A, 32 (1999), 6629-6642. doi: 10.1088/0305-4470/32/38/302.  Google Scholar

[14]

M. J. Gotay, J. Isenberg, J. E. Marsden and R. Montgomery, Momentum maps and classical relativistic fields I: Covariant Theory, arXiv:physics/9801019v2, (1999). Google Scholar

[15]

C. Günther, The polysymplectic Hamiltonian formalism in field theory and calculus of variations I: The local case, J. Differential Geom., 25 (1987), 23-53.  Google Scholar

[16]

F. Hélein and J. Kouneiher, Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage-Dedecker versus De Donder-Weyl, Adv. Theor. Math. Phys., 8 (2004), 565-601.  Google Scholar

[17]

I. V. Kanatchikov, Canonical structure of classical field theory in the polymomentum phase space, Rep. Math. Phys., 41 (1998), 49-90. doi: 10.1016/S0034-4877(98)80182-1.  Google Scholar

[18]

J. Kijowski, A finite-dimensional canonical formalism in the classical field theory, Comm. Math. Phys., 30 (1973), 99-128. doi: 10.1007/BF01645975.  Google Scholar

[19]

J. Kijowski and W. Szczyrba, Multisymplectic manifolds and the geometrical construction of the Poisson brackets in the classical field theory, Géométrie Symplectique et Physique Mathématique Coll. Int. C.N.R.S., 237 (1975), 347-378.  Google Scholar

[20]

J. Kijowski and W. M. Tulczyjew, "A Symplectic Framework for Field Theories," Lect. Notes Phys., 170, Springer-Verlag, Berlin, 1979.  Google Scholar

[21]

J. M. Lee, "Introduction to Smooth Manifolds,'' Springer, New York, 2003.  Google Scholar

[22]

M. de León, J. Marín-Solano and J. C. Marrero, A geometrical approach to classical field theories: A constraint algorithm for singular theories, Proc. on New Developments in Differential geometry, L. Tamassi-J. Szenthe eds., Kluwer Acad. Press, (1996), 291-312.  Google Scholar

[23]

M. de León, M. McLean, L. K. Norris, A. Rey-Roca and M. Salgado, Geometric structures in field theory, arXiv:math-ph/0208036v1 (2002). Google Scholar

[24]

M. de León, E. Merino, J. A. Oubiña, P. Rodrigues and M. Salgado, Hamiltonian systems on $k$-cosymplectic manifolds, J. Math. Phys., 39 (1998), 876-893.  Google Scholar

[25]

M. de León, E. Merino and M. Salgado, $k$-cosymplectic manifolds and Lagrangian field theories, J. Math. Phys., 42 (2001), 2092-2104.  Google Scholar

[26]

M. McLean and L. K. Norris, Covariant field theory on frame bundles of fibered manifolds, J. Math. Phys., 41 (2000), 6808-6823. doi: 10.1063/1.1288797.  Google Scholar

[27]

J. E. Marsden and S. Shkoller, Multisymplectic geometry, covariant Hamiltonians and water waves, Math. Proc. Camb. Phil. Soc., 125 (1999), 553-575. doi: 10.1017/S0305004198002953.  Google Scholar

[28]

F. Munteanu, A. M. Rey and M. Salgado, The Günther's formalism in classical field theory: Momentum map and reduction, J. Math. Phys., 45 (2004), 1730-1751. doi: 10.1063/1.1688433.  Google Scholar

[29]

M.C. Muñoz-Lecanda, M. Salgado and S. Vilariño, $k$-symplectic and $k$-cosymplectic Lagrangian field theories: Some interesting examples and applications, Int. J. Geom. Meth. Mod. Phys., 7 (2010), 669-692. doi: 10.1142/S0219887810004506.  Google Scholar

[30]

L. K. Norris, Generalized symplectic geometry on the frame bundle of a manifold, Proc. Symp. Pure Math. 54, Part 2. Amer. Math. Soc., Providence RI, (1993), 435-465.  Google Scholar

[31]

L. K. Norris, $n$-symplectic algebra of observables in covariant Lagrangian field theory, J. Math. Phys., 42 (2001), 4827-4845. doi: 10.1063/1.1396835.  Google Scholar

[32]

C. Paufler and H. Römer, Geometry of Hamiltonian $n$-vector fields in multisymplectic field theory, J. Geom. Phys., 44 (2002), 52-69.  Google Scholar

[33]

A. M. Rey, N. Román-Roy and M. Salgado, Günther's formalism in classical field theory: Skinner-Rusk approach and the evolution operator, J. Math. Phys., 46 (2005), 052901. doi: 10.1063/1.1876872.  Google Scholar

[34]

N. Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories, Symmetry Integrability Geom. Methods Appl. (SIGMA), 5 (2009) 100, 25 pp.  Google Scholar

[35]

N. Román-Roy, M. Salgado and S. Vilariño, On a kind of Noether symmetries and conservation laws in $k$-symplectic field theory, J. Math. Phys., 52 (2011), 022901; (20 pages). Google Scholar

[36]

G. Sardanashvily, "Generalized Hamiltonian Formalism for Field Theory. Constraint Systems,'' World Scientific, Singapore, 1995.  Google Scholar

[37]

D. J. Saunders, "The Geometry of Jet Bundles,'' London Math. Soc. Lect. Notes Ser. 142, Cambridge, Univ. Press, 1989.  Google Scholar

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