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1. | Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland |
2. | Division of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland, Poland |
References:
[1] |
C. Buttin, Théorie des opérateurs différentiels gradués sur les formes différentielles, Bull. Soc. Math. France, 102 (1974), 49-73. |
[2] |
F. Cantrijn, L. A. Ibort and M. De Leon, Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 225-236. |
[3] |
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), 1-26.
doi: 10.3934/jgm.2012.4.1. |
[4] |
J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order theories, Differential Geom. Appl., 1 (1991), 345-374.
doi: 10.1016/0926-2245(91)90013-Y. |
[5] |
A. Echeverría-Enríquez and M. C. Muñoz-Lecanda, Geometry of multisymplectic Hamiltonian first order theory, J. Math. Phys., 41 (2000), 7402-7444.
doi: 10.1063/1.1308075. |
[6] |
L. E. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 1998. |
[7] |
M. Forger, C. Paufler and H. Römer, A general construction of Poisson brackets on exact multisymplectic manifolds, Rep. Math. Phys., 51 (2003), 187-195.
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), 112903-112932.
doi: 10.1063/1.2116320. |
[9] |
M. Forger and L. G. Gomes, Multisymplectic and polysymplectic structures on fiber bundles, Rev. Math. Phys., 25 (2013), 1350018, 47pp.
doi: 10.1142/S0129055X13500189. |
[10] |
K. Gawędzki, On the geometrization of the canonical formalism in the classical field theory, Rep. Math. Phys., 3 (1972), 307-326.
doi: 10.1016/0034-4877(72)90014-6. |
[11] |
G. Giachetta and L. Mangiarotti, Constrained Hamiltonian Systems and Gauge Theories, Int. J. Theor. Phys., 34 (1995), 2353-2371.
doi: 10.1007/BF00670772. |
[12] |
G. Giachetta, L. Mangiarotti and G. A. Sardanashvili, Advanced Classical Field Theory, World Scientific, Singapore, 2009.
doi: 10.1142/9789812838964. |
[13] |
M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields, Part I: Covariant field theory, preprint, arXiv: physics/9801019. |
[14] |
M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields, Part II: Canonical analysis of field theories, preprint, arXiv: math-ph/0411032. |
[15] |
K. Grabowska, The Tulczyjew triple for classical fields, J. Phys. A, 45 (2012), 145207-145242.
doi: 10.1088/1751-8113/45/14/145207. |
[16] |
K. Grabowska, Lagrangian and Hamiltonian formalism in field theory: A simple model, J. Geom. Mech., 2 (2010), 375-395.
doi: 10.3934/jgm.2010.2.375. |
[17] |
K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204 (25pp).
doi: 10.1088/1751-8113/41/17/175204. |
[18] |
K. Grabowska and J. Grabowski, Dirac algebroids in Lagrangian and Hamiltonian mechanics, J. Geom. Phys., 61 (2011), 2233-2253.
doi: 10.1016/j.geomphys.2011.06.018. |
[19] |
K. Grabowska and J. Grabowski, Tulczyjew triples: From statics to field theory, J. Geom. Mech., 5 (2013), 445-472.
doi: 10.3934/jgm.2013.5.445. |
[20] |
K. Grabowska, J. Grabowski and P. Urbański, Geometrical Mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575.
doi: 10.1142/S0219887806001259. |
[21] |
J. Grabowski, Brakets, Int. J. Geom. Methods Mod. Phys. 10 (2013), 1360001, 45 pp.
doi: 10.1142/S0219887813600013. |
[22] |
J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305.
doi: 10.1016/j.geomphys.2009.06.009. |
[23] |
J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures, J.Geom. Phys., 62 (2012), 21-36.
doi: 10.1016/j.geomphys.2011.09.004. |
[24] |
J. Grabowski and P. Urbański, Tangent lifts of Poisson and related structures, J. Phys. A, 28 (1995), 6743-6777.
doi: 10.1088/0305-4470/28/23/024. |
[25] |
J. Grabowski and P. Urbański, Algebroids - general differential calculi on vector bundles, J. Geom. Phys., 31 (1999), 111-141.
doi: 10.1016/S0393-0440(99)00007-8. |
[26] |
C. Günther, The polysymplectic Hamiltonian formalism in field theory and calculus of variations. I. The local case, J. Differential Geom., 25 (1987), 23-53. |
[27] |
J. Kijowski and W. Szczyrba, A canonical structure for classical field theories, Commun. Math. Phys., 46 (1976), 183-206.
doi: 10.1007/BF01608496. |
[28] |
J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories, Lecture Notes in Physics, 107. Springer-Verlag, Berlin-New York, 1979. |
[29] |
I. Kolář and J. Tomáš, Gauge-natural transformations of some cotangent bundles, Acta Univ. M. Belii ser. Mathematics, 5 (1997), 3-9. |
[30] |
K. Konieczna and P. Urbański, Double vector bundles and duality, Arch. Math. (Brno), 35 (1999), 59-95. |
[31] |
O. Krupková, Hamiltonian field theory, J. Geom. Phys. 43 (2002), 93-132.
doi: 10.1016/S0393-0440(01)00087-0. |
[32] |
M. de León, D. Martín de Diego and A. Santamaría-Merino, Tulczyjew's triples and lagrangian submanifolds in classical field theories, in Applied Differential Geometry and Mechanics (eds. W. Sarlet and F. Cantrijn), Univ. of Gent, Gent, Academia Press, (2003), 21-47. |
[33] |
P. Libermann and C. M. Marle, Symplectic Geometry and Analytical Mechanics, Translated from the French by Bertram Eugene Schwarzbach. Mathematics and its Applications, 35. D. Reidel Publishing Co., Dordrecht, 1987.
doi: 10.1007/978-94-009-3807-6. |
[34] |
D. Lüst and S. Theisen, Lectures on String Theory, Lecture Notes in Physics, 346, Springer-Verlag, Berlin, 1989. |
[35] |
M. Łukasik, Rachunek Wariacyjny Niezale.zny od Parametryzacji. Przypadek Jednowymiarowy (Polish), PhD Thesis, University of Warsaw, 2012. |
[36] |
G. Martin, A Darboux theorem for multi-symplectic manifolds, Lett. Math. Phys., 16 (1988), 133-138.
doi: 10.1007/BF00402020. |
[37] |
G. Pidello and W. Tulczyjew, Derivations of differential forms on jet bundles, Ann. Mat. Pura Appl., 147 (1987), 249-265.
doi: 10.1007/BF01762420. |
[38] |
J. Pradines, Représentation des jets non holonomes par des morphismes vectoriels doubles soudés, C. R. Acad. Sci. Paris, série A, 278 (1974), 1523-1526. |
[39] |
D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in Quantization, Poisson brackets and beyond (Manchester, 2001),Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002, pp. 169-185.
doi: 10.1090/conm/315/05479. |
[40] |
G. Sardanashvily, Lagrangian dynamics of submanifolds. Relativistic mechanics, J. Geom. Mech., 4 (2012), 99-110.
doi: 10.3934/jgm.2012.4.99. |
[41] |
P. Ševera, Some title containing the words "homotopy" and "symplectic", e.g. this one, Travaux mathématiques, Univ. Luxemb., 16 (2005), 121-137. |
[42] |
W. Tulczyjew, Hamiltonian systems, Lagrangian systems, and the Legendre transformation, Symposia Math., 14 (1974), 247-258. |
[43] |
W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114. |
[44] |
W. M. Tulczyjew, A symplectic framework for linear field theories, Ann. Mat. Pura Appl., 130 (1982), 177-195.
doi: 10.1007/BF01761494. |
[45] |
W. M. Tulczyjew, Geometric Formulation of Physical Theories, Bibliopolis, Naples, 1989. |
[46] |
W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians, The Infeld Centennial Meeting (Warsaw,1998), Acta Phys. Polon. B, 30 (1999), 2909-2978. |
[47] |
L. Vitagliano, Partial differential Hamiltonian systems, Cand. J. Math., 65 (2013), 1164-1200.
doi: 10.4153/CJM-2012-055-0. |
[48] |
P. Urbański, Double vector bundles in classical mechanics, Rend. Sem. Matem. Torino, 54 (1996), 405-421. |
[49] |
T. T. Voronov, Graded manifolds and Drienfeld doubles for Lie bialgebroids, in Quantization, Poisson brackets and beyond (Manchester, 2001), Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002, pp. 131-168.
doi: 10.1090/conm/315/05478. |
[50] |
Y. Xin, Minimal Submanifolds and Related Topics, Nankai Tracts in Mathematics 8, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
doi: 10.1142/9789812564382. |
show all references
References:
[1] |
C. Buttin, Théorie des opérateurs différentiels gradués sur les formes différentielles, Bull. Soc. Math. France, 102 (1974), 49-73. |
[2] |
F. Cantrijn, L. A. Ibort and M. De Leon, Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 225-236. |
[3] |
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), 1-26.
doi: 10.3934/jgm.2012.4.1. |
[4] |
J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order theories, Differential Geom. Appl., 1 (1991), 345-374.
doi: 10.1016/0926-2245(91)90013-Y. |
[5] |
A. Echeverría-Enríquez and M. C. Muñoz-Lecanda, Geometry of multisymplectic Hamiltonian first order theory, J. Math. Phys., 41 (2000), 7402-7444.
doi: 10.1063/1.1308075. |
[6] |
L. E. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 1998. |
[7] |
M. Forger, C. Paufler and H. Römer, A general construction of Poisson brackets on exact multisymplectic manifolds, Rep. Math. Phys., 51 (2003), 187-195.
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), 112903-112932.
doi: 10.1063/1.2116320. |
[9] |
M. Forger and L. G. Gomes, Multisymplectic and polysymplectic structures on fiber bundles, Rev. Math. Phys., 25 (2013), 1350018, 47pp.
doi: 10.1142/S0129055X13500189. |
[10] |
K. Gawędzki, On the geometrization of the canonical formalism in the classical field theory, Rep. Math. Phys., 3 (1972), 307-326.
doi: 10.1016/0034-4877(72)90014-6. |
[11] |
G. Giachetta and L. Mangiarotti, Constrained Hamiltonian Systems and Gauge Theories, Int. J. Theor. Phys., 34 (1995), 2353-2371.
doi: 10.1007/BF00670772. |
[12] |
G. Giachetta, L. Mangiarotti and G. A. Sardanashvili, Advanced Classical Field Theory, World Scientific, Singapore, 2009.
doi: 10.1142/9789812838964. |
[13] |
M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields, Part I: Covariant field theory, preprint, arXiv: physics/9801019. |
[14] |
M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields, Part II: Canonical analysis of field theories, preprint, arXiv: math-ph/0411032. |
[15] |
K. Grabowska, The Tulczyjew triple for classical fields, J. Phys. A, 45 (2012), 145207-145242.
doi: 10.1088/1751-8113/45/14/145207. |
[16] |
K. Grabowska, Lagrangian and Hamiltonian formalism in field theory: A simple model, J. Geom. Mech., 2 (2010), 375-395.
doi: 10.3934/jgm.2010.2.375. |
[17] |
K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204 (25pp).
doi: 10.1088/1751-8113/41/17/175204. |
[18] |
K. Grabowska and J. Grabowski, Dirac algebroids in Lagrangian and Hamiltonian mechanics, J. Geom. Phys., 61 (2011), 2233-2253.
doi: 10.1016/j.geomphys.2011.06.018. |
[19] |
K. Grabowska and J. Grabowski, Tulczyjew triples: From statics to field theory, J. Geom. Mech., 5 (2013), 445-472.
doi: 10.3934/jgm.2013.5.445. |
[20] |
K. Grabowska, J. Grabowski and P. Urbański, Geometrical Mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575.
doi: 10.1142/S0219887806001259. |
[21] |
J. Grabowski, Brakets, Int. J. Geom. Methods Mod. Phys. 10 (2013), 1360001, 45 pp.
doi: 10.1142/S0219887813600013. |
[22] |
J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305.
doi: 10.1016/j.geomphys.2009.06.009. |
[23] |
J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures, J.Geom. Phys., 62 (2012), 21-36.
doi: 10.1016/j.geomphys.2011.09.004. |
[24] |
J. Grabowski and P. Urbański, Tangent lifts of Poisson and related structures, J. Phys. A, 28 (1995), 6743-6777.
doi: 10.1088/0305-4470/28/23/024. |
[25] |
J. Grabowski and P. Urbański, Algebroids - general differential calculi on vector bundles, J. Geom. Phys., 31 (1999), 111-141.
doi: 10.1016/S0393-0440(99)00007-8. |
[26] |
C. Günther, The polysymplectic Hamiltonian formalism in field theory and calculus of variations. I. The local case, J. Differential Geom., 25 (1987), 23-53. |
[27] |
J. Kijowski and W. Szczyrba, A canonical structure for classical field theories, Commun. Math. Phys., 46 (1976), 183-206.
doi: 10.1007/BF01608496. |
[28] |
J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories, Lecture Notes in Physics, 107. Springer-Verlag, Berlin-New York, 1979. |
[29] |
I. Kolář and J. Tomáš, Gauge-natural transformations of some cotangent bundles, Acta Univ. M. Belii ser. Mathematics, 5 (1997), 3-9. |
[30] |
K. Konieczna and P. Urbański, Double vector bundles and duality, Arch. Math. (Brno), 35 (1999), 59-95. |
[31] |
O. Krupková, Hamiltonian field theory, J. Geom. Phys. 43 (2002), 93-132.
doi: 10.1016/S0393-0440(01)00087-0. |
[32] |
M. de León, D. Martín de Diego and A. Santamaría-Merino, Tulczyjew's triples and lagrangian submanifolds in classical field theories, in Applied Differential Geometry and Mechanics (eds. W. Sarlet and F. Cantrijn), Univ. of Gent, Gent, Academia Press, (2003), 21-47. |
[33] |
P. Libermann and C. M. Marle, Symplectic Geometry and Analytical Mechanics, Translated from the French by Bertram Eugene Schwarzbach. Mathematics and its Applications, 35. D. Reidel Publishing Co., Dordrecht, 1987.
doi: 10.1007/978-94-009-3807-6. |
[34] |
D. Lüst and S. Theisen, Lectures on String Theory, Lecture Notes in Physics, 346, Springer-Verlag, Berlin, 1989. |
[35] |
M. Łukasik, Rachunek Wariacyjny Niezale.zny od Parametryzacji. Przypadek Jednowymiarowy (Polish), PhD Thesis, University of Warsaw, 2012. |
[36] |
G. Martin, A Darboux theorem for multi-symplectic manifolds, Lett. Math. Phys., 16 (1988), 133-138.
doi: 10.1007/BF00402020. |
[37] |
G. Pidello and W. Tulczyjew, Derivations of differential forms on jet bundles, Ann. Mat. Pura Appl., 147 (1987), 249-265.
doi: 10.1007/BF01762420. |
[38] |
J. Pradines, Représentation des jets non holonomes par des morphismes vectoriels doubles soudés, C. R. Acad. Sci. Paris, série A, 278 (1974), 1523-1526. |
[39] |
D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in Quantization, Poisson brackets and beyond (Manchester, 2001),Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002, pp. 169-185.
doi: 10.1090/conm/315/05479. |
[40] |
G. Sardanashvily, Lagrangian dynamics of submanifolds. Relativistic mechanics, J. Geom. Mech., 4 (2012), 99-110.
doi: 10.3934/jgm.2012.4.99. |
[41] |
P. Ševera, Some title containing the words "homotopy" and "symplectic", e.g. this one, Travaux mathématiques, Univ. Luxemb., 16 (2005), 121-137. |
[42] |
W. Tulczyjew, Hamiltonian systems, Lagrangian systems, and the Legendre transformation, Symposia Math., 14 (1974), 247-258. |
[43] |
W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114. |
[44] |
W. M. Tulczyjew, A symplectic framework for linear field theories, Ann. Mat. Pura Appl., 130 (1982), 177-195.
doi: 10.1007/BF01761494. |
[45] |
W. M. Tulczyjew, Geometric Formulation of Physical Theories, Bibliopolis, Naples, 1989. |
[46] |
W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians, The Infeld Centennial Meeting (Warsaw,1998), Acta Phys. Polon. B, 30 (1999), 2909-2978. |
[47] |
L. Vitagliano, Partial differential Hamiltonian systems, Cand. J. Math., 65 (2013), 1164-1200.
doi: 10.4153/CJM-2012-055-0. |
[48] |
P. Urbański, Double vector bundles in classical mechanics, Rend. Sem. Matem. Torino, 54 (1996), 405-421. |
[49] |
T. T. Voronov, Graded manifolds and Drienfeld doubles for Lie bialgebroids, in Quantization, Poisson brackets and beyond (Manchester, 2001), Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002, pp. 131-168.
doi: 10.1090/conm/315/05478. |
[50] |
Y. Xin, Minimal Submanifolds and Related Topics, Nankai Tracts in Mathematics 8, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
doi: 10.1142/9789812564382. |
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