Tulczyjew triples in higher derivative field theory

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March 2015, 7(1): 1-33. doi: 10.3934/jgm.2015.7.1

Tulczyjew triples in higher derivative field theory

Katarzyna Grabowska ^1, and Luca Vitagliano ^2,

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

Received September 2014 Revised December 2014 Published March 2015

Abstract
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The geometrical structure known as Tulczyjew triple has been used with success in analytical mechanics and first order field theory to describe a wide range of physical systems, including Lagrangian/Hamiltonian systems with constraints and/or sources, or with singular Lagrangian. Starting from the first principles of the variational calculus, we derive Tulczyjew triples for classical field theories of arbitrary high order, i.e. depending on arbitrarily high derivatives of the fields. A first triple appears as the result of considering higher order theories as first order ones with configurations being constrained to be holonomic jets. A second triple is obtained after a reduction procedure aimed at getting rid of nonphysical degrees of freedom. This picture we present is fully covariant and complete: it contains both Lagrangian and Hamiltonian formalisms, in particular the Euler-Lagrange equations. Notice that the required Geometry of jet bundles is affine (as opposed to the linear Geometry of the tangent bundle). Accordingly, the notions of affine duality and affine phase space play a distinguished role in our picture. In particular the Tulczyjew triples in this paper consist of morphisms of double affine-vector bundles which, moreover, preserve suitable presymplectic structures.

Keywords: Tulczyjew triple, jet spaces., Lagrangian formalism, Hamiltonian formalism, classical field theory, variational calculus.

Mathematics Subject Classification: Primary: 70S05, 70H03, 70H05; Secondary: 58A20, 53D0.

Citation: Katarzyna Grabowska, Luca Vitagliano. Tulczyjew triples in higher derivative field theory. Journal of Geometric Mechanics, 2015, 7 (1) : 1-33. doi: 10.3934/jgm.2015.7.1

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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