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, (1978).

[2]

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

[3]

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

[4]

A. Awane and M. Goze, "Pfaffian Systems, $k$-Symplectic Systems,'', Kluwer Acad. Pub., (2000).

[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.

[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.

[7]

J. Dieudonné, "Foundations of Modern Analysis,'', 2nd ed., (1969).

[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).

[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.

[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.

[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.

[12]

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

[13]

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

[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).

[15]

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

[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.

[17]

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

[18]

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

[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.

[20]

J. Kijowski and W. M. Tulczyjew, "A Symplectic Framework for Field Theories,", Lect. Notes Phys., 170 (1979).

[21]

J. M. Lee, "Introduction to Smooth Manifolds,'', Springer, (2003).

[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, (1996), 291.

[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)., (2002).

[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.

[25]

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

[26]

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

[27]

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

[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. doi: 10.1063/1.1688433.

[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. doi: 10.1142/S0219887810004506.

[30]

L. K. Norris, Generalized symplectic geometry on the frame bundle of a manifold,, Proc. Symp. Pure Math. {\bf 54}, 54 (1993), 435.

[31]

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

[32]

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

[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). doi: 10.1063/1.1876872.

[34]

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

[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).

[36]

G. Sardanashvily, "Generalized Hamiltonian Formalism for Field Theory. Constraint Systems,'', World Scientific, (1995).

[37]

D. J. Saunders, "The Geometry of Jet Bundles,'', London Math. Soc. Lect. Notes Ser. {\bf 142}, 142 (1989).

show all references

References:
[1]

R. A. Abraham and J. E. Marsden, "Foundations of Mechanics,'', 2nd Edition, (1978).

[2]

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

[3]

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

[4]

A. Awane and M. Goze, "Pfaffian Systems, $k$-Symplectic Systems,'', Kluwer Acad. Pub., (2000).

[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.

[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.

[7]

J. Dieudonné, "Foundations of Modern Analysis,'', 2nd ed., (1969).

[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).

[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.

[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.

[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.

[12]

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

[13]

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

[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).

[15]

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

[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.

[17]

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

[18]

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

[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.

[20]

J. Kijowski and W. M. Tulczyjew, "A Symplectic Framework for Field Theories,", Lect. Notes Phys., 170 (1979).

[21]

J. M. Lee, "Introduction to Smooth Manifolds,'', Springer, (2003).

[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, (1996), 291.

[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)., (2002).

[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.

[25]

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

[26]

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

[27]

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

[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. doi: 10.1063/1.1688433.

[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. doi: 10.1142/S0219887810004506.

[30]

L. K. Norris, Generalized symplectic geometry on the frame bundle of a manifold,, Proc. Symp. Pure Math. {\bf 54}, 54 (1993), 435.

[31]

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

[32]

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

[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). doi: 10.1063/1.1876872.

[34]

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

[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).

[36]

G. Sardanashvily, "Generalized Hamiltonian Formalism for Field Theory. Constraint Systems,'', World Scientific, (1995).

[37]

D. J. Saunders, "The Geometry of Jet Bundles,'', London Math. Soc. Lect. Notes Ser. {\bf 142}, 142 (1989).

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