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).   Google Scholar

[2]

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

[3]

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

[4]

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

[7]

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

[12]

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

[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.  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.   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.   Google Scholar

[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.  Google Scholar

[18]

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

[20]

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

[21]

J. M. Lee, "Introduction to Smooth Manifolds,'', Springer, (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, (1996), 291.   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)., (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.   Google Scholar

[25]

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

[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.  Google Scholar

[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.  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.  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.  doi: 10.1142/S0219887810004506.  Google Scholar

[30]

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

[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.  Google Scholar

[32]

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

[36]

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

[37]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[7]

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

[12]

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

[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.  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.   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.   Google Scholar

[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.  Google Scholar

[18]

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

[20]

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

[21]

J. M. Lee, "Introduction to Smooth Manifolds,'', Springer, (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, (1996), 291.   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)., (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.   Google Scholar

[25]

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

[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.  Google Scholar

[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.  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.  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.  doi: 10.1142/S0219887810004506.  Google Scholar

[30]

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

[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.  Google Scholar

[32]

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

[36]

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

[37]

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

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