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.

[2]

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

[3]

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

[4]

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

[5]

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

[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.

[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.

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[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.

[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.

[27]

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

[28]

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

[29]

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

[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.

[31]

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

[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).

[33]

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

[34]

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

[35]

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

[36]

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

[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.

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[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.

[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.

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[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.

[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.

[27]

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

[28]

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

[29]

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

[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.

[31]

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

[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).

[33]

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

[34]

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

[35]

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

[36]

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

[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.

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