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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[26]

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

[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.

[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.

[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.

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[26]

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

[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.

[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.

[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.

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

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