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

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

Received  September 2014 Revised  December 2014 Published  March 2015

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.
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, (2011). doi: 10.1007/978-1-4614-1499-5. Google Scholar

[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, (1999). Google Scholar

[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). doi: 10.1088/1751-8113/42/47/475207. Google Scholar

[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). doi: 10.1088/1751-8113/47/23/235203. Google Scholar

[5]

T. de Donder, Théorie Invariantive du Calcul des Variations,, Gauthier-Villars, (1935). Google Scholar

[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), (2003), 21. Google Scholar

[7]

K. Grabowska, Lagrangian and Hamiltonian formalism in Field Theory: A simple model,, J. Geom. Mech., 2 (2010), 375. doi: 10.3934/jgm.2010.2.375. Google Scholar

[8]

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

[9]

K. Grabowska and P. Urbański, AV-differential geometry and Newtonian Mechanics,, Rep. Math. Phys., 58 (2006), 21. doi: 10.1016/S0034-4877(06)80038-8. Google Scholar

[10]

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

[11]

K. Grabowska, J. Grabowski and P. Urbański, AV-differential geometry: Euler-Lagrange equations,, J. Geom. Phys., 57 (2007), 1984. doi: 10.1016/j.geomphys.2007.04.003. Google Scholar

[12]

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

[13]

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

[14]

J. Kijowski, Elasticità Finita e Relativistica: Introduzione ai Metodi Geometrici della Teoria dei Campi,, Quaderni dell'Unione Matematica Italiana, (1991). Google Scholar

[15]

J. Kijowski and G. Moreno, Symplectic structures related with higher order variational problems,, e-print, (). Google Scholar

[16]

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

[17]

P. Liebermann and Ch. M. Marle, Symplectic Geometry and Analytical Mechanics,, Reidel Publishing Company, (1987). doi: 10.1007/978-94-009-3807-6. Google Scholar

[18]

D. J. Saunders, The Geometry of Jet Bundles,, Cambridge University Press, (1989). doi: 10.1017/CBO9780511526411. Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

W. M. Tulczyjew, The Euler-Lagrange resolution,, in Differential Geometrical Methods in Mathematical Physics, (1980), 22. doi: 10.1007/BFb0089725. Google Scholar

[26]

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

[27]

W. M. Tulczyjew and P. Urbański, Liouville structures,, Universitatis Iagellonicae Acta Mathematica, 47 (2009), 187. Google Scholar

[28]

A. M. Vinogradov, The $\mathcalC$-spectral sequence, Lagrangian formalism and conservation laws I. The linear theory,, J. Math. Anal. Appl., 100 (1984), 1. doi: 10.1016/0022-247X(84)90071-4. Google Scholar

[29]

A. M. Vinogradov, The $\mathcalC$-spectral sequence, Lagrangian formalism and conservation laws II. The nonlinear theory,, J. Math. Anal. Appl., 100 (1984), 41. doi: 10.1016/0022-247X(84)90072-6. Google Scholar

[30]

L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher order field theories,, J. Geom. Phys., 60 (2010), 857. doi: 10.1016/j.geomphys.2010.02.003. Google Scholar

[31]

L. Vitagliano, The Hamilton-Jacobi formalism for higher order field theories,, Int. J. Geom. Meth. Mod. Phys., 7 (2010), 1413. doi: 10.1142/S0219887810004889. Google Scholar

[32]

L. Vitagliano, Geometric Hamilton-Jacobi field theory,, Int. J. Geom. Meth. Mod. Phys., 9 (2012). doi: 10.1142/S0219887812600080. Google Scholar

show all references

References:
[1]

S. Benenti, Hamiltonian Structures and Generating Families,, Universitext, (2011). doi: 10.1007/978-1-4614-1499-5. Google Scholar

[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, (1999). Google Scholar

[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). doi: 10.1088/1751-8113/42/47/475207. Google Scholar

[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). doi: 10.1088/1751-8113/47/23/235203. Google Scholar

[5]

T. de Donder, Théorie Invariantive du Calcul des Variations,, Gauthier-Villars, (1935). Google Scholar

[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), (2003), 21. Google Scholar

[7]

K. Grabowska, Lagrangian and Hamiltonian formalism in Field Theory: A simple model,, J. Geom. Mech., 2 (2010), 375. doi: 10.3934/jgm.2010.2.375. Google Scholar

[8]

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

[9]

K. Grabowska and P. Urbański, AV-differential geometry and Newtonian Mechanics,, Rep. Math. Phys., 58 (2006), 21. doi: 10.1016/S0034-4877(06)80038-8. Google Scholar

[10]

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

[11]

K. Grabowska, J. Grabowski and P. Urbański, AV-differential geometry: Euler-Lagrange equations,, J. Geom. Phys., 57 (2007), 1984. doi: 10.1016/j.geomphys.2007.04.003. Google Scholar

[12]

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

[13]

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

[14]

J. Kijowski, Elasticità Finita e Relativistica: Introduzione ai Metodi Geometrici della Teoria dei Campi,, Quaderni dell'Unione Matematica Italiana, (1991). Google Scholar

[15]

J. Kijowski and G. Moreno, Symplectic structures related with higher order variational problems,, e-print, (). Google Scholar

[16]

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

[17]

P. Liebermann and Ch. M. Marle, Symplectic Geometry and Analytical Mechanics,, Reidel Publishing Company, (1987). doi: 10.1007/978-94-009-3807-6. Google Scholar

[18]

D. J. Saunders, The Geometry of Jet Bundles,, Cambridge University Press, (1989). doi: 10.1017/CBO9780511526411. Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

W. M. Tulczyjew, The Euler-Lagrange resolution,, in Differential Geometrical Methods in Mathematical Physics, (1980), 22. doi: 10.1007/BFb0089725. Google Scholar

[26]

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

[27]

W. M. Tulczyjew and P. Urbański, Liouville structures,, Universitatis Iagellonicae Acta Mathematica, 47 (2009), 187. Google Scholar

[28]

A. M. Vinogradov, The $\mathcalC$-spectral sequence, Lagrangian formalism and conservation laws I. The linear theory,, J. Math. Anal. Appl., 100 (1984), 1. doi: 10.1016/0022-247X(84)90071-4. Google Scholar

[29]

A. M. Vinogradov, The $\mathcalC$-spectral sequence, Lagrangian formalism and conservation laws II. The nonlinear theory,, J. Math. Anal. Appl., 100 (1984), 41. doi: 10.1016/0022-247X(84)90072-6. Google Scholar

[30]

L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher order field theories,, J. Geom. Phys., 60 (2010), 857. doi: 10.1016/j.geomphys.2010.02.003. Google Scholar

[31]

L. Vitagliano, The Hamilton-Jacobi formalism for higher order field theories,, Int. J. Geom. Meth. Mod. Phys., 7 (2010), 1413. doi: 10.1142/S0219887810004889. Google Scholar

[32]

L. Vitagliano, Geometric Hamilton-Jacobi field theory,, Int. J. Geom. Meth. Mod. Phys., 9 (2012). doi: 10.1142/S0219887812600080. Google Scholar

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