December  2010, 2(4): 375-395. doi: 10.3934/jgm.2010.2.375

Lagrangian and Hamiltonian formalism in Field Theory: A simple model

1. 

Faculty of Physics, University of Warsaw, Hoza 69, 00-681 Warszawa, Poland

Received  July 2010 Revised  November 2010 Published  January 2011

The static of smooth maps from the two-dimensional disc to a smooth manifold can be regarded as a simplified version of the Classical Field Theory. In this paper we construct the Tulczyjew triple for the problem and describe the Lagrangian and Hamiltonian formalism. We outline also natural generalizations of this approach to arbitrary dimensions.
Citation: 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
References:
[1]

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

[2]

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

[3]

F. Cantrijn, L. A. Ibort and M. De Leon, Hamiltonian structures on multisymplectic manifolds,, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996).   Google Scholar

[4]

J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order theories,, Differential Geom. Appl., 1 (1991), 354.   Google Scholar

[5]

J. F. Cariñena, M. Crampin and L. A. Ibort, On the geometry of multisymplectic manifolds,, J. Austral. Math. Soc. A, 66 (1999), 303.   Google Scholar

[6]

A. Echeverria-Enriquez and M. C. Muñoz-Lecanda, Geometry of multisymplectic Hamiltonian first order theory,, J. Math. Phys., 41 (2000), 7402.  doi: 10.1063/1.1308075.  Google Scholar

[7]

M. Forger, C. Paufler and H. Römer, A general construction of Poisson brackets on exact multisymplectic manifolds,, Rep. Math. Phys., 51 (2003), 187.  doi: 10.1016/S0034-4877(03)80012-5.  Google Scholar

[8]

M. Forger, C. Paufler and H. Römer, Hamiltonian multivector fields and Poisson forms in multisymplectic field theories,, J Math. Phys., 46 (2005).   Google Scholar

[9]

G. Giachetta and L. Mangiarotti, Constrained Hamiltonian systems and Gauge theories,, Int. J. Theor. Phys., 34 (1995), 2353.  doi: 10.1007/BF00670772.  Google Scholar

[10]

M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields, Part I: Covariant field theory,, , (2004).   Google Scholar

[11]

M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields, Part II: Canonical analysis of field theories,, , (2004).   Google Scholar

[12]

M. J. Gotay, A multisymplecitc framework for classical field theory and the calculus of variations I: Covariant Hamiltonian formalism,, in, (1991).   Google Scholar

[13]

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.  doi: 10.1016/0926-2245(91)90014-Z.  Google Scholar

[14]

J. Grabowski and P. Urbański, Algebroids - general differential calculi on vector bundles,, J. Geom. Phys., 31 (1999), 111.  doi: 10.1016/S0393-0440(99)00007-8.  Google Scholar

[15]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids,, J. Phys. A: Math. Theor., 41 (2008).   Google Scholar

[16]

K. Grabowska, J. Grabowski and P. Urbanski, AV-differential geometry: Poisson and Jacobi structures,, J. Geom. Phys., 52 (2004), 398.  doi: 10.1016/j.geomphys.2004.04.004.  Google Scholar

[17]

K. Grabowska, J. Grabowski and P. Urbanski, Geometrical mechanics on algebroids,, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559.  doi: 10.1142/S0219887806001259.  Google Scholar

[18]

T. Gotō, Relativistic quantum mechanics of one-dimensional mechanical continuum and subsidiary condition of dual resonance model,, Prog. Theor. Phys., 46 (1971), 1560.  doi: 10.1143/PTP.46.1560.  Google Scholar

[19]

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

[20]

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

[21]

J. Kijowski, Elasticità finita e relativistica: introduzione ai metodi geometrici della teoria dei campi,, Pitagora Editrice (Bologna) (1991)., (1991).   Google Scholar

[22]

J. Kijowski and W. M. Tulczyjew, "A Symplectic Framework for Field Theories,", Lecture Notes in Physics, 107 (1979).   Google Scholar

[23]

J. Klein, Espaces variationelles et mécanique,, Ann. Inst. Fourier (Grenoble), 12 (1962), 1.   Google Scholar

[24]

K. Konieczna and P. Urbański, Double vector bundles and duality,, Arch. Math. (Brno), 35 (1999), 59.   Google Scholar

[25]

M. de León, J.-C. Marrero, E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids,, J. Phys. A: Math. Gen., 38 (2005), 241.  doi: 10.1088/0305-4470/38/24/R01.  Google Scholar

[26]

M. de León, D. Martín de Diego and A. Santamaría-Merino, Tulczyjew’s triples and lagrangian submanifolds in classical field theories,, in, (2003), 21.   Google Scholar

[27]

E. Martínez, Lagrangian mechanics on Lie algebroids,, Acta Appl. Math., 67 (2001), 295.  doi: 10.1023/A:1011965919259.  Google Scholar

[28]

E. Martínez, Variational calculus on Lie algebroids,, ESAIM Control Optim. Calc. Var., 14 (2008), 356.  doi: 10.1051/cocv:2007056.  Google Scholar

[29]

Y. Nambu, "Lectures prepared for the Copenhagen Summer Symposium,", (unpublished) (1970)., (1970).   Google Scholar

[30]

A. De Nicola and W. M. Tulczyjew, A variational formulation of electrodynamics with external sources,, Int. J. Geom. Methods Mod. Phys., 6 (2009), 173.  doi: 10.1142/S0219887809003461.  Google Scholar

[31]

A. M. Polyakov, Quantum geometry of bosonic strings,, Phys. Lett. B, 103 (1981), 207.  doi: 10.1016/0370-2693(81)90743-7.  Google Scholar

[32]

A. M. Rey, N. Roman-Roy, M. Salgado and S. Vilariño, k-Cosymplectic classical field theories: Tulczyjew, Skinner-Rusk and Lie algebroid formulations,, , (2008).   Google Scholar

[33]

W. M. Tulczyjew, The origin of variational principles in Classical and quantum integrabilty,, (Warsaw, 59 (2003), 41.   Google Scholar

[34]

W. M. Tulczyjew, Hamiltonian systems, Lagrangian systems, and the Legendre transformation,, Symposia Mathematica, (1974).   Google Scholar

[35]

W. M. Tulczyjew, "Geometric Formulation of Physical Theories,", Bibliopolis, (1989).   Google Scholar

[36]

W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians,, The Infeld Centennial Meeting (Warsaw, 30 (1999), 2909.   Google Scholar

[37]

J. Vankershaver, F. Cantrijn, M. De Leon and M. De Diego, Geometric aspects of nonholonomic field theories,, Rep. Math. Phys., 46 (2005), 387.  doi: 10.1016/S0034-4877(05)80093-X.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

F. Cantrijn, L. A. Ibort and M. De Leon, Hamiltonian structures on multisymplectic manifolds,, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996).   Google Scholar

[4]

J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order theories,, Differential Geom. Appl., 1 (1991), 354.   Google Scholar

[5]

J. F. Cariñena, M. Crampin and L. A. Ibort, On the geometry of multisymplectic manifolds,, J. Austral. Math. Soc. A, 66 (1999), 303.   Google Scholar

[6]

A. Echeverria-Enriquez and M. C. Muñoz-Lecanda, Geometry of multisymplectic Hamiltonian first order theory,, J. Math. Phys., 41 (2000), 7402.  doi: 10.1063/1.1308075.  Google Scholar

[7]

M. Forger, C. Paufler and H. Römer, A general construction of Poisson brackets on exact multisymplectic manifolds,, Rep. Math. Phys., 51 (2003), 187.  doi: 10.1016/S0034-4877(03)80012-5.  Google Scholar

[8]

M. Forger, C. Paufler and H. Römer, Hamiltonian multivector fields and Poisson forms in multisymplectic field theories,, J Math. Phys., 46 (2005).   Google Scholar

[9]

G. Giachetta and L. Mangiarotti, Constrained Hamiltonian systems and Gauge theories,, Int. J. Theor. Phys., 34 (1995), 2353.  doi: 10.1007/BF00670772.  Google Scholar

[10]

M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields, Part I: Covariant field theory,, , (2004).   Google Scholar

[11]

M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields, Part II: Canonical analysis of field theories,, , (2004).   Google Scholar

[12]

M. J. Gotay, A multisymplecitc framework for classical field theory and the calculus of variations I: Covariant Hamiltonian formalism,, in, (1991).   Google Scholar

[13]

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.  doi: 10.1016/0926-2245(91)90014-Z.  Google Scholar

[14]

J. Grabowski and P. Urbański, Algebroids - general differential calculi on vector bundles,, J. Geom. Phys., 31 (1999), 111.  doi: 10.1016/S0393-0440(99)00007-8.  Google Scholar

[15]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids,, J. Phys. A: Math. Theor., 41 (2008).   Google Scholar

[16]

K. Grabowska, J. Grabowski and P. Urbanski, AV-differential geometry: Poisson and Jacobi structures,, J. Geom. Phys., 52 (2004), 398.  doi: 10.1016/j.geomphys.2004.04.004.  Google Scholar

[17]

K. Grabowska, J. Grabowski and P. Urbanski, Geometrical mechanics on algebroids,, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559.  doi: 10.1142/S0219887806001259.  Google Scholar

[18]

T. Gotō, Relativistic quantum mechanics of one-dimensional mechanical continuum and subsidiary condition of dual resonance model,, Prog. Theor. Phys., 46 (1971), 1560.  doi: 10.1143/PTP.46.1560.  Google Scholar

[19]

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

[20]

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

[21]

J. Kijowski, Elasticità finita e relativistica: introduzione ai metodi geometrici della teoria dei campi,, Pitagora Editrice (Bologna) (1991)., (1991).   Google Scholar

[22]

J. Kijowski and W. M. Tulczyjew, "A Symplectic Framework for Field Theories,", Lecture Notes in Physics, 107 (1979).   Google Scholar

[23]

J. Klein, Espaces variationelles et mécanique,, Ann. Inst. Fourier (Grenoble), 12 (1962), 1.   Google Scholar

[24]

K. Konieczna and P. Urbański, Double vector bundles and duality,, Arch. Math. (Brno), 35 (1999), 59.   Google Scholar

[25]

M. de León, J.-C. Marrero, E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids,, J. Phys. A: Math. Gen., 38 (2005), 241.  doi: 10.1088/0305-4470/38/24/R01.  Google Scholar

[26]

M. de León, D. Martín de Diego and A. Santamaría-Merino, Tulczyjew’s triples and lagrangian submanifolds in classical field theories,, in, (2003), 21.   Google Scholar

[27]

E. Martínez, Lagrangian mechanics on Lie algebroids,, Acta Appl. Math., 67 (2001), 295.  doi: 10.1023/A:1011965919259.  Google Scholar

[28]

E. Martínez, Variational calculus on Lie algebroids,, ESAIM Control Optim. Calc. Var., 14 (2008), 356.  doi: 10.1051/cocv:2007056.  Google Scholar

[29]

Y. Nambu, "Lectures prepared for the Copenhagen Summer Symposium,", (unpublished) (1970)., (1970).   Google Scholar

[30]

A. De Nicola and W. M. Tulczyjew, A variational formulation of electrodynamics with external sources,, Int. J. Geom. Methods Mod. Phys., 6 (2009), 173.  doi: 10.1142/S0219887809003461.  Google Scholar

[31]

A. M. Polyakov, Quantum geometry of bosonic strings,, Phys. Lett. B, 103 (1981), 207.  doi: 10.1016/0370-2693(81)90743-7.  Google Scholar

[32]

A. M. Rey, N. Roman-Roy, M. Salgado and S. Vilariño, k-Cosymplectic classical field theories: Tulczyjew, Skinner-Rusk and Lie algebroid formulations,, , (2008).   Google Scholar

[33]

W. M. Tulczyjew, The origin of variational principles in Classical and quantum integrabilty,, (Warsaw, 59 (2003), 41.   Google Scholar

[34]

W. M. Tulczyjew, Hamiltonian systems, Lagrangian systems, and the Legendre transformation,, Symposia Mathematica, (1974).   Google Scholar

[35]

W. M. Tulczyjew, "Geometric Formulation of Physical Theories,", Bibliopolis, (1989).   Google Scholar

[36]

W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians,, The Infeld Centennial Meeting (Warsaw, 30 (1999), 2909.   Google Scholar

[37]

J. Vankershaver, F. Cantrijn, M. De Leon and M. De Diego, Geometric aspects of nonholonomic field theories,, Rep. Math. Phys., 46 (2005), 387.  doi: 10.1016/S0034-4877(05)80093-X.  Google Scholar

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