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Tulczyjew triples in higher derivative field theory
1. | Department of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland |
2. | Department of Mathematics, University of Salerno, and Istituto Nazionale di Fisica Nucleare, GC Salerno, Via Giovanni Paolo II, 123, 84084 Fisciano (SA), Italy |
References:
[1] |
S. Benenti, Hamiltonian Structures and Generating Families, Universitext, Springer, New York, 2011.
doi: 10.1007/978-1-4614-1499-5. |
[2] |
A. V. Bocharov, et al., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Edited and with a preface by I. S. Krasil'shchik and A. M. Vinogradov, Transl. Math. Mon., 182, Amer. Math. Soc., Providence, 1999. |
[3] |
C. M. Campos, M. de León, D. Martín de Diego and J. Vankerschaver, Unambiguous formalism for higher order Lagrangian field theories, J. Phys. A: Math. Theor., 42 (2009), 475207 (24pp).
doi: 10.1088/1751-8113/42/47/475207. |
[4] |
L. Colombo, M. de León, P. D. Prieto-Martínez and N. Román-Roy, Geometric Hamilton-Jacobi theory for higher-order autonomous systems, J. Phys. A: Math. Theor., 47 (2014), 235203 (24pp).
doi: 10.1088/1751-8113/47/23/235203. |
[5] |
T. de Donder, Théorie Invariantive du Calcul des Variations, Gauthier-Villars, Paris, 1935. |
[6] |
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), Academia Press, Gent, 2003, 21-47. |
[7] |
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. |
[8] |
K. Grabowska, A Tulczyjew triple for classical fields, J. Phys. A: Math. Theor., 45 (2012), 145207 (35pp).
doi: 10.1088/1751-8113/45/14/145207. |
[9] |
K. Grabowska and P. Urbański, AV-differential geometry and Newtonian Mechanics, Rep. Math. Phys., 58 (2006), 21-40.
doi: 10.1016/S0034-4877(06)80038-8. |
[10] |
K. Grabowska, J. Grabowski and P. Urbański, AV-differential geometry: Poisson and Jacobi structures, J. Geom. Phys., 52 (2004), 398-446.
doi: 10.1016/j.geomphys.2004.04.004. |
[11] |
K. Grabowska, J. Grabowski and P. Urbański, AV-differential geometry: Euler-Lagrange equations, J. Geom. Phys., 57 (2007), 1984-1998.
doi: 10.1016/j.geomphys.2007.04.003. |
[12] |
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. |
[13] |
J. Grabowski, M. Rotkiewicz and P. Urbański, Double affine bundles, J. Geom. Phys., 60 (2010), 581-598.
doi: 10.1016/j.geomphys.2009.12.008. |
[14] |
J. Kijowski, Elasticità Finita e Relativistica: Introduzione ai Metodi Geometrici della Teoria dei Campi, Quaderni dell'Unione Matematica Italiana, 37, Pitagora Editrice, Bologna, 1991. |
[15] |
J. Kijowski and G. Moreno, Symplectic structures related with higher order variational problems,, e-print, ().
|
[16] |
K. Konieczna and P. Urbański, Double vector bundles and duality, Arch. Math. (Brno), 35 (1999), 59-95. |
[17] |
P. Liebermann and Ch. M. Marle, Symplectic Geometry and Analytical Mechanics, Reidel Publishing Company, Dordrecht, 1987.
doi: 10.1007/978-94-009-3807-6. |
[18] |
D. J. Saunders, The Geometry of Jet Bundles, Cambridge University Press, Cambridge, 1989.
doi: 10.1017/CBO9780511526411. |
[19] |
W. M. Tulczyjew, Geometric Formulation of Physical Theories. Statics and Dynamics of Mechanical Systems, Monographs and Textbooks in Physical Science, Lecture Notes, 11, Bibliopolis, Naples, 1989. |
[20] |
W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114. |
[21] |
W. M. Tulczyjew, Relations symplectiques et les équations d'Hamilton-Jacobi relativistes, C. R. Acad. Sc. Paris Sér., 281 (1975), A545-A547. |
[22] |
W. M. Tulczyjew, A symplectic framework of linear field theories, Ann. Mat. Pura Appl., 130 (1982), 177-195.
doi: 10.1007/BF01761494. |
[23] |
W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sc. Paris, 283 (1976), A15-A18. |
[24] |
W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sc. Paris, 283 (1976), A675-A678. |
[25] |
W. M. Tulczyjew, The Euler-Lagrange resolution, in Differential Geometrical Methods in Mathematical Physics, Lect. Notes Math., 836, Springer, Berlin, 1980, 22-48.
doi: 10.1007/BFb0089725. |
[26] |
W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians, Acta Phys. Polon. B, 30 (1999), 2909-2978. |
[27] |
W. M. Tulczyjew and P. Urbański, Liouville structures, Universitatis Iagellonicae Acta Mathematica, 47 (2009), 187-226. |
[28] |
A. M. Vinogradov, The $\mathcalC$-spectral sequence, Lagrangian formalism and conservation laws I. The linear theory, J. Math. Anal. Appl., 100 (1984), 1-40.
doi: 10.1016/0022-247X(84)90071-4. |
[29] |
A. M. Vinogradov, The $\mathcalC$-spectral sequence, Lagrangian formalism and conservation laws II. The nonlinear theory, J. Math. Anal. Appl., 100 (1984), 41-129.
doi: 10.1016/0022-247X(84)90072-6. |
[30] |
L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher order field theories, J. Geom. Phys., 60 (2010), 857-873.
doi: 10.1016/j.geomphys.2010.02.003. |
[31] |
L. Vitagliano, The Hamilton-Jacobi formalism for higher order field theories, Int. J. Geom. Meth. Mod. Phys., 7 (2010), 1413-1436.
doi: 10.1142/S0219887810004889. |
[32] |
L. Vitagliano, Geometric Hamilton-Jacobi field theory, Int. J. Geom. Meth. Mod. Phys., 9 (2012), 1260008 (8pp).
doi: 10.1142/S0219887812600080. |
show all references
References:
[1] |
S. Benenti, Hamiltonian Structures and Generating Families, Universitext, Springer, New York, 2011.
doi: 10.1007/978-1-4614-1499-5. |
[2] |
A. V. Bocharov, et al., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Edited and with a preface by I. S. Krasil'shchik and A. M. Vinogradov, Transl. Math. Mon., 182, Amer. Math. Soc., Providence, 1999. |
[3] |
C. M. Campos, M. de León, D. Martín de Diego and J. Vankerschaver, Unambiguous formalism for higher order Lagrangian field theories, J. Phys. A: Math. Theor., 42 (2009), 475207 (24pp).
doi: 10.1088/1751-8113/42/47/475207. |
[4] |
L. Colombo, M. de León, P. D. Prieto-Martínez and N. Román-Roy, Geometric Hamilton-Jacobi theory for higher-order autonomous systems, J. Phys. A: Math. Theor., 47 (2014), 235203 (24pp).
doi: 10.1088/1751-8113/47/23/235203. |
[5] |
T. de Donder, Théorie Invariantive du Calcul des Variations, Gauthier-Villars, Paris, 1935. |
[6] |
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), Academia Press, Gent, 2003, 21-47. |
[7] |
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. |
[8] |
K. Grabowska, A Tulczyjew triple for classical fields, J. Phys. A: Math. Theor., 45 (2012), 145207 (35pp).
doi: 10.1088/1751-8113/45/14/145207. |
[9] |
K. Grabowska and P. Urbański, AV-differential geometry and Newtonian Mechanics, Rep. Math. Phys., 58 (2006), 21-40.
doi: 10.1016/S0034-4877(06)80038-8. |
[10] |
K. Grabowska, J. Grabowski and P. Urbański, AV-differential geometry: Poisson and Jacobi structures, J. Geom. Phys., 52 (2004), 398-446.
doi: 10.1016/j.geomphys.2004.04.004. |
[11] |
K. Grabowska, J. Grabowski and P. Urbański, AV-differential geometry: Euler-Lagrange equations, J. Geom. Phys., 57 (2007), 1984-1998.
doi: 10.1016/j.geomphys.2007.04.003. |
[12] |
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. |
[13] |
J. Grabowski, M. Rotkiewicz and P. Urbański, Double affine bundles, J. Geom. Phys., 60 (2010), 581-598.
doi: 10.1016/j.geomphys.2009.12.008. |
[14] |
J. Kijowski, Elasticità Finita e Relativistica: Introduzione ai Metodi Geometrici della Teoria dei Campi, Quaderni dell'Unione Matematica Italiana, 37, Pitagora Editrice, Bologna, 1991. |
[15] |
J. Kijowski and G. Moreno, Symplectic structures related with higher order variational problems,, e-print, ().
|
[16] |
K. Konieczna and P. Urbański, Double vector bundles and duality, Arch. Math. (Brno), 35 (1999), 59-95. |
[17] |
P. Liebermann and Ch. M. Marle, Symplectic Geometry and Analytical Mechanics, Reidel Publishing Company, Dordrecht, 1987.
doi: 10.1007/978-94-009-3807-6. |
[18] |
D. J. Saunders, The Geometry of Jet Bundles, Cambridge University Press, Cambridge, 1989.
doi: 10.1017/CBO9780511526411. |
[19] |
W. M. Tulczyjew, Geometric Formulation of Physical Theories. Statics and Dynamics of Mechanical Systems, Monographs and Textbooks in Physical Science, Lecture Notes, 11, Bibliopolis, Naples, 1989. |
[20] |
W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114. |
[21] |
W. M. Tulczyjew, Relations symplectiques et les équations d'Hamilton-Jacobi relativistes, C. R. Acad. Sc. Paris Sér., 281 (1975), A545-A547. |
[22] |
W. M. Tulczyjew, A symplectic framework of linear field theories, Ann. Mat. Pura Appl., 130 (1982), 177-195.
doi: 10.1007/BF01761494. |
[23] |
W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sc. Paris, 283 (1976), A15-A18. |
[24] |
W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sc. Paris, 283 (1976), A675-A678. |
[25] |
W. M. Tulczyjew, The Euler-Lagrange resolution, in Differential Geometrical Methods in Mathematical Physics, Lect. Notes Math., 836, Springer, Berlin, 1980, 22-48.
doi: 10.1007/BFb0089725. |
[26] |
W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians, Acta Phys. Polon. B, 30 (1999), 2909-2978. |
[27] |
W. M. Tulczyjew and P. Urbański, Liouville structures, Universitatis Iagellonicae Acta Mathematica, 47 (2009), 187-226. |
[28] |
A. M. Vinogradov, The $\mathcalC$-spectral sequence, Lagrangian formalism and conservation laws I. The linear theory, J. Math. Anal. Appl., 100 (1984), 1-40.
doi: 10.1016/0022-247X(84)90071-4. |
[29] |
A. M. Vinogradov, The $\mathcalC$-spectral sequence, Lagrangian formalism and conservation laws II. The nonlinear theory, J. Math. Anal. Appl., 100 (1984), 41-129.
doi: 10.1016/0022-247X(84)90072-6. |
[30] |
L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher order field theories, J. Geom. Phys., 60 (2010), 857-873.
doi: 10.1016/j.geomphys.2010.02.003. |
[31] |
L. Vitagliano, The Hamilton-Jacobi formalism for higher order field theories, Int. J. Geom. Meth. Mod. Phys., 7 (2010), 1413-1436.
doi: 10.1142/S0219887810004889. |
[32] |
L. Vitagliano, Geometric Hamilton-Jacobi field theory, Int. J. Geom. Meth. Mod. Phys., 9 (2012), 1260008 (8pp).
doi: 10.1142/S0219887812600080. |
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