December  2013, 5(4): 445-472. doi: 10.3934/jgm.2013.5.445

Tulczyjew triples: From statics to field theory

1. 

Physics Department, University of Warsaw, Hoża 69, 00-681 Warszawa, Poland

2. 

Polish Academy of Sciences, Institute of Mathematics, Śniadeckich 8, P.O. Box 21, 00-956 Warsaw

Received  April 2013 Revised  September 2013 Published  December 2013

A geometric approach to dynamical equations of physics, based on the idea of the Tulczyjew triple, is presented. We show the evolution of these concepts, starting with the roots lying in the variational calculus for statics, through Lagrangian and Hamiltonian mechanics, and ending with Tulczyjew triples for classical field theories illustrated with a few important examples.
Citation: Katarzyna Grabowska, Janusz Grabowski. Tulczyjew triples: From statics to field theory. Journal of Geometric Mechanics, 2013, 5 (4) : 445-472. doi: 10.3934/jgm.2013.5.445
References:
[1]

F. Cantrijn, L. A. Ibort and M. de León, Hamiltonian structures on multisymplectic manifolds,, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 225.   Google Scholar

[2]

C. M. Campos, E. Guzmán and J. C. Marrero, Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds,, J. Geom. Mech., 4 (2012), 1.  doi: 10.3934/jgm.2012.4.1.  Google Scholar

[3]

J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order theories,, Differential Geom. Appl., 1 (1991), 354.  doi: 10.1016/0926-2245(91)90013-Y.  Google Scholar

[4]

J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids,, IMA J. Math. Control. Inform., 21 (2004), 457.  doi: 10.1093/imamci/21.4.457.  Google Scholar

[5]

A. Echeverría-Enríquez 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

[6]

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

[7]

M. Forger, C. Paufler and H. Römer, Hamiltonian multivector fields and Poisson forms in multisymplectic field theories,, J. Math. Phys., 46 (2005).  doi: 10.1063/1.2116320.  Google Scholar

[8]

K. Gawędzki, On the geometrization of the canonical formalism in the classical field theory,, Rep. Math. Phys., 3 (1972), 307.  doi: 10.1016/0034-4877(72)90014-6.  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]

G. Giachetta, L. Mangiarotti and G. A. Sardanashvili, Advanced Classical Field Theory,, World Scientific Publishing Co. Pte. Ltd., (2009).  doi: 10.1142/9789812838964.  Google Scholar

[11]

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

[12]

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

[13]

K. Grabowska, A Tulczyjew triple for classical fields,, J. Phys. A, 45 (2012).  doi: 10.1088/1751-8113/45/14/145207.  Google Scholar

[14]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids,, J. Phys. A, 41 (2008).  doi: 10.1088/1751-8113/41/17/175204.  Google Scholar

[15]

K. Grabowska and J. Grabowski, Dirac algebroids in Lagrangian and Hamiltonian mechanics,, J. Geom. Phys., 61 (2011), 2233.  doi: 10.1016/j.geomphys.2011.06.018.  Google Scholar

[16]

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

[17]

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

[18]

J. Grabowski and M. Jóźwikowski, Pontryagin maximum principle on almost Lie algebroids,, SIAM J. Control Optim., 49 (2011), 1306.  doi: 10.1137/090760246.  Google Scholar

[19]

J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds,, J. Geom. Phys., 59 (2009), 1285.  doi: 10.1016/j.geomphys.2009.06.009.  Google Scholar

[20]

J. Grabowski, M. Rotkiewicz and P. Urbański, Double affine bundles,, J. Geom. Phys., 60 (2010), 581.  doi: 10.1016/j.geomphys.2009.12.008.  Google Scholar

[21]

J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures,, J. Geom. Phys., 62 (2012), 21.  doi: 10.1016/j.geomphys.2011.09.004.  Google Scholar

[22]

J. Grabowski and P. Urbański, Tangent lifts of Poisson and related structures,, J. Phys. A, 28 (1995), 6743.  doi: 10.1088/0305-4470/28/23/024.  Google Scholar

[23]

J. Grabowski and P. Urbański, Lie algebroids and Poisson-Nijenhuis structures,, Rep. Math. Phys., 40 (1997), 195.  doi: 10.1016/S0034-4877(97)85916-2.  Google Scholar

[24]

J. Kijowski and W. Szczyrba, A canonical structure for classical field theories,, Commun. Math. Phys., 46 (1976), 183.  doi: 10.1007/BF01608496.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

M. de León, D. Martín de Diego and A. Santamaría-Merino, Tulczyjew triples and Lagrangian submanifolds in classical field theories,, in Applied Differential Geometry and Mechanics (eds. W. Sarlet and F. Cantrijn), (2003), 21.   Google Scholar

[29]

P. Liebermann and C. M. Marle, Symplectic Geometry and Analytical Mechanics,, Mathematics and its Applications, (1987).  doi: 10.1007/978-94-009-3807-6.  Google Scholar

[30]

K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras,, Bull. London Math. Soc., 27 (1995), 97.  doi: 10.1112/blms/27.2.97.  Google Scholar

[31]

J. Pradines, Fibrés Vectoriels Doubles et Calcul des Jets Non Holonomes,, (French) [Double Vector Bundles and the Calculus of Nonholonomic Jets], (1977).   Google Scholar

[32]

W. M. Tulczyjew, Geometric Formulation of Physical Theories. Statics and Dynamics of Mechanical Systems,, Monographs and Textbooks in Physical Science. Lecture Notes, (1989).   Google Scholar

[33]

W. M. Tulczyjew, The Legendre transformation,, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101.   Google Scholar

[34]

W. M. Tulczyjew, Relations symplectiques et les équations d'Hamilton-Jacobi relativistes,, (French) C. R. Acad. Sc. Paris Sér. A-B, 281 (1975).   Google Scholar

[35]

W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne,, (French) C. R. Acad. Sc. Paris Sér. A-B, 283 (1976).   Google Scholar

[36]

W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne,, (French) C. R. Acad. Sc. Paris Sér. A-B, 283 (1976).   Google Scholar

[37]

W. M. Tulczyjew, A symplectic framework for linear field theories,, Ann. Mat. Pura Appl. (4), 130 (1982), 177.  doi: 10.1007/BF01761494.  Google Scholar

[38]

W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians,, Acta Phys. Polon. B, 30 (1999), 2909.   Google Scholar

[39]

W. M. Tulczyjew and P. Urbański, Liouville structures,, Univ. Iagel. Acta Math., 47 (2009), 187.   Google Scholar

[40]

L. Vitagliano, Partial differential Hamiltonian systems,, Cand. J. Math., 65 (2013), 1164.  doi: 10.4153/CJM-2012-055-0.  Google Scholar

show all references

References:
[1]

F. Cantrijn, L. A. Ibort and M. de León, Hamiltonian structures on multisymplectic manifolds,, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 225.   Google Scholar

[2]

C. M. Campos, E. Guzmán and J. C. Marrero, Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds,, J. Geom. Mech., 4 (2012), 1.  doi: 10.3934/jgm.2012.4.1.  Google Scholar

[3]

J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order theories,, Differential Geom. Appl., 1 (1991), 354.  doi: 10.1016/0926-2245(91)90013-Y.  Google Scholar

[4]

J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids,, IMA J. Math. Control. Inform., 21 (2004), 457.  doi: 10.1093/imamci/21.4.457.  Google Scholar

[5]

A. Echeverría-Enríquez 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

[6]

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

[7]

M. Forger, C. Paufler and H. Römer, Hamiltonian multivector fields and Poisson forms in multisymplectic field theories,, J. Math. Phys., 46 (2005).  doi: 10.1063/1.2116320.  Google Scholar

[8]

K. Gawędzki, On the geometrization of the canonical formalism in the classical field theory,, Rep. Math. Phys., 3 (1972), 307.  doi: 10.1016/0034-4877(72)90014-6.  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]

G. Giachetta, L. Mangiarotti and G. A. Sardanashvili, Advanced Classical Field Theory,, World Scientific Publishing Co. Pte. Ltd., (2009).  doi: 10.1142/9789812838964.  Google Scholar

[11]

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

[12]

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

[13]

K. Grabowska, A Tulczyjew triple for classical fields,, J. Phys. A, 45 (2012).  doi: 10.1088/1751-8113/45/14/145207.  Google Scholar

[14]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids,, J. Phys. A, 41 (2008).  doi: 10.1088/1751-8113/41/17/175204.  Google Scholar

[15]

K. Grabowska and J. Grabowski, Dirac algebroids in Lagrangian and Hamiltonian mechanics,, J. Geom. Phys., 61 (2011), 2233.  doi: 10.1016/j.geomphys.2011.06.018.  Google Scholar

[16]

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

[17]

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

[18]

J. Grabowski and M. Jóźwikowski, Pontryagin maximum principle on almost Lie algebroids,, SIAM J. Control Optim., 49 (2011), 1306.  doi: 10.1137/090760246.  Google Scholar

[19]

J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds,, J. Geom. Phys., 59 (2009), 1285.  doi: 10.1016/j.geomphys.2009.06.009.  Google Scholar

[20]

J. Grabowski, M. Rotkiewicz and P. Urbański, Double affine bundles,, J. Geom. Phys., 60 (2010), 581.  doi: 10.1016/j.geomphys.2009.12.008.  Google Scholar

[21]

J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures,, J. Geom. Phys., 62 (2012), 21.  doi: 10.1016/j.geomphys.2011.09.004.  Google Scholar

[22]

J. Grabowski and P. Urbański, Tangent lifts of Poisson and related structures,, J. Phys. A, 28 (1995), 6743.  doi: 10.1088/0305-4470/28/23/024.  Google Scholar

[23]

J. Grabowski and P. Urbański, Lie algebroids and Poisson-Nijenhuis structures,, Rep. Math. Phys., 40 (1997), 195.  doi: 10.1016/S0034-4877(97)85916-2.  Google Scholar

[24]

J. Kijowski and W. Szczyrba, A canonical structure for classical field theories,, Commun. Math. Phys., 46 (1976), 183.  doi: 10.1007/BF01608496.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

M. de León, D. Martín de Diego and A. Santamaría-Merino, Tulczyjew triples and Lagrangian submanifolds in classical field theories,, in Applied Differential Geometry and Mechanics (eds. W. Sarlet and F. Cantrijn), (2003), 21.   Google Scholar

[29]

P. Liebermann and C. M. Marle, Symplectic Geometry and Analytical Mechanics,, Mathematics and its Applications, (1987).  doi: 10.1007/978-94-009-3807-6.  Google Scholar

[30]

K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras,, Bull. London Math. Soc., 27 (1995), 97.  doi: 10.1112/blms/27.2.97.  Google Scholar

[31]

J. Pradines, Fibrés Vectoriels Doubles et Calcul des Jets Non Holonomes,, (French) [Double Vector Bundles and the Calculus of Nonholonomic Jets], (1977).   Google Scholar

[32]

W. M. Tulczyjew, Geometric Formulation of Physical Theories. Statics and Dynamics of Mechanical Systems,, Monographs and Textbooks in Physical Science. Lecture Notes, (1989).   Google Scholar

[33]

W. M. Tulczyjew, The Legendre transformation,, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101.   Google Scholar

[34]

W. M. Tulczyjew, Relations symplectiques et les équations d'Hamilton-Jacobi relativistes,, (French) C. R. Acad. Sc. Paris Sér. A-B, 281 (1975).   Google Scholar

[35]

W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne,, (French) C. R. Acad. Sc. Paris Sér. A-B, 283 (1976).   Google Scholar

[36]

W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne,, (French) C. R. Acad. Sc. Paris Sér. A-B, 283 (1976).   Google Scholar

[37]

W. M. Tulczyjew, A symplectic framework for linear field theories,, Ann. Mat. Pura Appl. (4), 130 (1982), 177.  doi: 10.1007/BF01761494.  Google Scholar

[38]

W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians,, Acta Phys. Polon. B, 30 (1999), 2909.   Google Scholar

[39]

W. M. Tulczyjew and P. Urbański, Liouville structures,, Univ. Iagel. Acta Math., 47 (2009), 187.   Google Scholar

[40]

L. Vitagliano, Partial differential Hamiltonian systems,, Cand. J. Math., 65 (2013), 1164.  doi: 10.4153/CJM-2012-055-0.  Google Scholar

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