A new multisymplectic unified formalism for second order classical field theories

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June 2015, 7(2): 203-253. doi: 10.3934/jgm.2015.7.203

A new multisymplectic unified formalism for second order classical field theories

Pedro Daniel Prieto-Martínez ^1, and Narciso Román-Roy ^2,

1.	Departamento de Matemática Aplicada IV, Universitat Politècnica de Catalunya, Campus Norte, Ed. C3. C/ Jordi Girona 1, E-08034 Barcelona, Spain
2.	Departamento de Matemática Aplicada IV, Universitat Politècnica de Catalunya-BarcelonaTech, Campus Norte, Ed. C-3. C/ Jordi Girona 1, E-08034 Barcelona

Received February 2014 Revised March 2015 Published June 2015

Abstract
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We present a new multisymplectic framework for second-order classical field theories which is based on an extension of the unified Lagrangian-Hamiltonian formalism to these kinds of systems. This model provides a straightforward and simple way to define the Poincaré-Cartan form and clarifies the construction of the Legendre map (univocally obtained as a consequence of the constraint algorithm). Likewise, it removes the undesirable arbitrariness in the solutions to the field equations, which are analyzed in-depth, and written in terms of holonomic sections and multivector fields. Our treatment therefore completes previous attempt to achieve this aim. The formulation is applied to describing some physical examples; in particular, to giving another alternative multisymplectic description of the Korteweg-de Vries equation.

Keywords: multisymplectic manifolds, Lagrangian and Hamiltonian formalisms, Skinner-Rusk formulation, Higher-order field theories, KdV equation..

Mathematics Subject Classification: Primary: 70H50, 70S05, 35G99; Secondary: 53D42, 55R1.

Citation: Pedro Daniel Prieto-Martínez, Narciso Román-Roy. A new multisymplectic unified formalism for second order classical field theories. Journal of Geometric Mechanics, 2015, 7 (2) : 203-253. doi: 10.3934/jgm.2015.7.203

References:

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics, $2^{nd}$ edition, Benjamin-Cummings, New York, 1978.
[2]	V. Aldaya and J. A. de Azcarraga, Variational principles on $r-th$ order jets of fibre bundles in field theory, J. Math. Phys., 19 (1978), 1869-1875. doi: 10.1063/1.523904.
[3]	V. Aldaya and J. A. de Azcarraga, Vector Bundles, $r-th$ order Noether invariant and canonical symmetries in Lagrangian field theory, J. Math. Phys., 19 (1978), 1876-1880. doi: 10.1063/1.523905.
[4]	V. Aldaya and J. A. de Azcarraga, Higher order Hamiltonian formalism in field theory, J. Phys. A, 13 (1980), 2545-2551. doi: 10.1088/0305-4470/13/8/004.
[5]	U. M. Ascher and R. I. McLachlan, On symplectic and multisymplectic schemes for the KdV equation, J. Sci. Comput., 25 (2005), 83-104. doi: 10.1007/s10915-004-4634-6.
[6]	M. Barbero-Liñán, A. Echeverría-Enríquez, D. Martín de Diego, M. C. Muñoz-Lecanda and N. Román-Roy, Skinner-rusk unified formalism for optimal control systems and applications, J. Math. Phys. A: Math. Theor., 40 (2007), 12071-12093. doi: 10.1088/1751-8113/40/40/005.
[7]	M. Barbero-Liñán, A. Echeverría-Enríquez, D. Martín de Diego, M. C. Muñoz-Lecanda and N. Román-Roy, Unified formalism for non-autonomous mechanical systems, J. Math. Phys., 49 (2008), 062902, 14pp. doi: 10.1063/1.2929668.
[8]	C. Batlle, J. Gomis, J. M. Pons and N. Román-Roy, Lagrangian and Hamiltonian constraints for second-order singular Lagrangians, J. Phys. A: Math. Gen., 21 (1988), 2693-2703. doi: 10.1088/0305-4470/21/12/013.
[9]	C. M. Campos, Geometric Methods in Classical Field Theory and Continous Media, Ph.D. thesis, Universidad Autónoma de Madrid, 2010.
[10]	C. M. Campos, Higher-order field theory with constraints, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106 (2012), 89-95. doi: 10.1007/s13398-011-0025-7.
[11]	C. M. Campos, M. de León, D. Martín de Diego and J. Vankerschaver, Unambigous formalism for higher-order lagrangian field theories, J. Phys A: Math Theor., 42 (2009), 475207, 24pp. doi: 10.1088/1751-8113/42/47/475207.
[12]	F. Cantrijn, M. Crampin and W. Sarlet, Higher-order differential equations and higher-order Lagrangian mechanics, Math. Proc. Cambridge Philos. Soc., 99 (1986), 565-587. doi: 10.1017/S0305004100064501.
[13]	J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order field theories, Diff. Geom. Appl., 1 (1991), 345-374. doi: 10.1016/0926-2245(91)90013-Y.
[14]	L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach, J. Math. Phys., 51 (2010), 083519, 24pp. doi: 10.1063/1.3456158.
[15]	J. Cortés, S. Martínez and F. Cantrijn, Skinner-Rusk approach to time-dependent mechanics, Physics Letters A, 300 (2002), 250-258. doi: 10.1016/S0375-9601(02)00777-6.
[16]	M. Crampin and D. J. Saunders, Homogeneity and projective equivalence of differential equation fields, J. Geom. Mech., 4 (2012), 27-47. doi: 10.3934/jgm.2012.4.27.
[17]	M. de León, J. Marín-Solano and J. C. Marrero, The constraint algorithm in the jet formalism, Nuovo Cimento B, (11) 84 (1984), 91-101.
[18]	M. de León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz-Lecanda and N. Román-Roy, Premultisymplectic constraint algorithm for field theories, Int. J. Geom. Methods Mod. Phys., 2 (2005), 839-871.
[19]	M. de León, J. C. Marrero and D. Martín de Diego, A new geometric setting for classical field theories, Banach Center Publ., 59 (2003), 189-209. doi: 10.4064/bc59-0-10.
[20]	M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory, North-Holland Math. Studies, 112, Elsevier Science Publishers B.V., Amsterdam, 1985.
[21]	M. de León and P. R. Rodrigues, Higher-order almost tangent geometry and non-autonomous Lagrangian dynamics, Rend. Circ. Mat. Palermo, 2 (1987), 157-171.
[22]	A. Echeverría-Enríquez, M. De León, M. C. Muñoz-Lecanda and N. Román-Roy, Extended Hamiltonian systems in multisymplectic field theories, J. Math. Phys., 48 (2007), 112901, 30pp. doi: 10.1063/1.2801875.
[23]	A. Echeverría-Enríquez, C. López, J. Marín-Solano, M. C. Muñoz-Lecanda and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for field theory, J. Math. Phys., 45 (2004), 360-380. doi: 10.1063/1.1628384.
[24]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector fields and connections: Setting Lagrangian equations in field theories, J. Math. Phys., 39 (1998), 4578-4603. doi: 10.1063/1.532525.
[25]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys., 41 (2000), 7402-7444. doi: 10.1063/1.1308075.
[26]	M. Ferraris and M. Francaviglia, Applications of the Poincaré-Cartan form in higher order field theories, in Differential Geometry and its Applications (Brno, 1986), Math. Appl. (East European Ser.) 27, Reidel, Dordrecht, 1987, 31-52.
[27]	M. Francaviglia and D. Krupka, The Hamiltonian formalism in higher order variational problems, Ann. Inst. H. Poincaré Sect. A (N.S.), 37 (1982), 295-315.
[28]	P. L. García and J. Muñoz-Masqué, On the geometrical structure of higher order variational calculus, in Procs. IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, Vol. I (Torino, 1982), Atti. Accad. Sci. Torino Cl. Sci. Fis. Math. Natur. 117, 1983, 127-147.
[29]	P. L. García and J. Muñoz-Masqué, Higher order regular variational problems, in Symplectic Geometry and Mathematical Physics (Aix-en-Provence, 1990), Progr. Math., 99, Birkhäuser Boston, MA, 1991, 136-159.
[30]	M. J. Gotay, A multisymplectic approach to the KdV equation, in Diferential Geometric Methods in Theoretical Physics (eds., K. Bleuler and M. Werner), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 250, Kluwer Acad. Publ., Dordrecht, 1988, 295-305.
[31]	K. Grabowska and L. Vitagliano, Tulczyjew triples in higher derivative field theory, J. Geom. Mech., 7 (2015), 1-33. doi: 10.3934/jgm.2015.7.1.
[32]	X. Gràcia, J. M. Pons and N. Román-Roy, Higher-order Lagrangian systems: Geometric structures, dynamics and constraints, J. Math. Phys., 32 (1991), 2744-2763. doi: 10.1063/1.529066.
[33]	X. Gràcia, J. M. Pons and N. Román-Roy, Higher-order conditions for singular Lagrangian systems, J. Phys. A: Math. Gen., 25 (1992), 1989-2004. doi: 10.1088/0305-4470/25/7/037.
[34]	D. R. Grigore, On a generalization of the Poincaré-Cartan form in higher-order field theory, in Variations, Geometry and Physics, Nova Sci. Publ., New York, 2009, 57-76.
[35]	M. Horák and I. Kolár, On the higher order Poincaré-Cartan forms, Czechoslovak Math. J., 33 (1983), 467-475.
[36]	I. Kolár, A geometrical version of the higher order Hamilton formalism in fibered manifolds, J. Geom. and Phys., 1 (1984), 127-137. doi: 10.1016/0393-0440(84)90007-X.
[37]	S. Kouranbaeva and S. Shkoller, A variational approach to second-order multisymplectic field theory, J. Geom. Phys., 35 (2000), 333-366. doi: 10.1016/S0393-0440(00)00012-7.
[38]	D. Krupka, On the higher order Hamilton theory in fibered spaces, in Procs. Conference on Differential Geometry and its Applications, Part 2, Univ. J. E. Purkyně, Brno, 1984, 167-183.
[39]	O. Krupkova, Higher-order mechanical systems with constraints, J. Math. Phys., 41 (2000), 5304-5324. doi: 10.1063/1.533411.
[40]	R. Miron, The Geometry of Higher-order Hamilton Spaces: Applications to Hamiltonian Mechanics, Fundamental Theories of Physics, 132, Springer-Verlag, New York. 2003. doi: 10.1007/978-94-010-0070-3.
[41]	P. Mukherjee and B. Paul, Gauge invariances of higher derivative Maxwell-Chern-Simons field theories: A new Hamiltonian approach, Phys. Rev. D, 85 (2012), 045028. doi: 10.1103/PhysRevD.85.045028.
[42]	J. Muñoz-Masqué, Canonical Cartan equations for higher order variational problems, J. Geom. Phys., 1 (1984), 1-7. doi: 10.1016/0393-0440(84)90001-9.
[43]	J. Muñoz-Masqué, Poincaré-Cartan forms in higher order variational calculus on fibred manifolds, Rev. Mat. Iberoamericana, 1 (1985), 85-126. doi: 10.4171/RMI/20.
[44]	P. D. Prieto-Martínez and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems, J. Phys. A Math. Theor., 44 (2011), 385203, 35pp. doi: 10.1088/1751-8113/44/38/385203.
[45]	P. D. Prieto-Martínez, N. Román-Roy, Unified formalism for higher-order non-autonomous dynamical systems, J. Math. Phys., 53 (2012), 032901, 38pp. doi: 10.1063/1.3692326.
[46]	P. D. Prieto-Martínez and N. Román-Roy, Higher-order mechanics: Variational principles and other topics, J. Geom. Mech., 5 (2013), 493-510. doi: 10.3934/jgm.2013.5.493.
[47]	A. M. Rey, N. Román-Roy and M. Salgado, Gunther's formalism k-symplectic formalism in classical field theory: Skinner-Rusk approach and the evolution operator, J. Math. Phys., 46 (2005), 052901, 24pp. doi: 10.1063/1.1876872.
[48]	A. M. Rey, N. Román-Roy, M. Salgado and S. Vilariño, k-cosymplectic classical field theories: Tulckzyjew and Skinner-Rusk formulations, Math. Phys. Anal. Geom., 15 (2012), 85-119. doi: 10.1007/s11040-012-9104-z.
[49]	D. J. Saunders, An alternative approach to the Cartan form in Lagrangian field theories, J. Phys. A, 20 (1987), 339-349. doi: 10.1088/0305-4470/20/2/019.
[50]	D. J. Saunders, The Geometry of Jet Bundles, London Mathematical Society, Lecture Notes Series, 142, Cambridge University Press, Cambridge, New York, 1989. doi: 10.1017/CBO9780511526411.
[51]	D. J. Saunders and M. Crampin, On the Legendre map in higher-order field theories, J. Phys. A: Math. Gen., 23 (1990), 3169-3182. doi: 10.1088/0305-4470/23/14/016.
[52]	R. Skinner and R. Rusk, Generalized Hamiltonian dynamics. I. Formulation on $T^Q\oplus TQ$, J. Math. Phys.,* 24 (1983), 2589-2594. doi: 10.1063/1.525654.
[53]	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.
[54]	P. F. Zhao and M. Z. Qin, Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation, J. Phys. A, 33 (2000), 3613-3626. doi: 10.1088/0305-4470/33/18/308.

show all references

References:

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics, $2^{nd}$ edition, Benjamin-Cummings, New York, 1978.
[2]	V. Aldaya and J. A. de Azcarraga, Variational principles on $r-th$ order jets of fibre bundles in field theory, J. Math. Phys., 19 (1978), 1869-1875. doi: 10.1063/1.523904.
[3]	V. Aldaya and J. A. de Azcarraga, Vector Bundles, $r-th$ order Noether invariant and canonical symmetries in Lagrangian field theory, J. Math. Phys., 19 (1978), 1876-1880. doi: 10.1063/1.523905.
[4]	V. Aldaya and J. A. de Azcarraga, Higher order Hamiltonian formalism in field theory, J. Phys. A, 13 (1980), 2545-2551. doi: 10.1088/0305-4470/13/8/004.
[5]	U. M. Ascher and R. I. McLachlan, On symplectic and multisymplectic schemes for the KdV equation, J. Sci. Comput., 25 (2005), 83-104. doi: 10.1007/s10915-004-4634-6.
[6]	M. Barbero-Liñán, A. Echeverría-Enríquez, D. Martín de Diego, M. C. Muñoz-Lecanda and N. Román-Roy, Skinner-rusk unified formalism for optimal control systems and applications, J. Math. Phys. A: Math. Theor., 40 (2007), 12071-12093. doi: 10.1088/1751-8113/40/40/005.
[7]	M. Barbero-Liñán, A. Echeverría-Enríquez, D. Martín de Diego, M. C. Muñoz-Lecanda and N. Román-Roy, Unified formalism for non-autonomous mechanical systems, J. Math. Phys., 49 (2008), 062902, 14pp. doi: 10.1063/1.2929668.
[8]	C. Batlle, J. Gomis, J. M. Pons and N. Román-Roy, Lagrangian and Hamiltonian constraints for second-order singular Lagrangians, J. Phys. A: Math. Gen., 21 (1988), 2693-2703. doi: 10.1088/0305-4470/21/12/013.
[9]	C. M. Campos, Geometric Methods in Classical Field Theory and Continous Media, Ph.D. thesis, Universidad Autónoma de Madrid, 2010.
[10]	C. M. Campos, Higher-order field theory with constraints, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106 (2012), 89-95. doi: 10.1007/s13398-011-0025-7.
[11]	C. M. Campos, M. de León, D. Martín de Diego and J. Vankerschaver, Unambigous formalism for higher-order lagrangian field theories, J. Phys A: Math Theor., 42 (2009), 475207, 24pp. doi: 10.1088/1751-8113/42/47/475207.
[12]	F. Cantrijn, M. Crampin and W. Sarlet, Higher-order differential equations and higher-order Lagrangian mechanics, Math. Proc. Cambridge Philos. Soc., 99 (1986), 565-587. doi: 10.1017/S0305004100064501.
[13]	J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order field theories, Diff. Geom. Appl., 1 (1991), 345-374. doi: 10.1016/0926-2245(91)90013-Y.
[14]	L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach, J. Math. Phys., 51 (2010), 083519, 24pp. doi: 10.1063/1.3456158.
[15]	J. Cortés, S. Martínez and F. Cantrijn, Skinner-Rusk approach to time-dependent mechanics, Physics Letters A, 300 (2002), 250-258. doi: 10.1016/S0375-9601(02)00777-6.
[16]	M. Crampin and D. J. Saunders, Homogeneity and projective equivalence of differential equation fields, J. Geom. Mech., 4 (2012), 27-47. doi: 10.3934/jgm.2012.4.27.
[17]	M. de León, J. Marín-Solano and J. C. Marrero, The constraint algorithm in the jet formalism, Nuovo Cimento B, (11) 84 (1984), 91-101.
[18]	M. de León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz-Lecanda and N. Román-Roy, Premultisymplectic constraint algorithm for field theories, Int. J. Geom. Methods Mod. Phys., 2 (2005), 839-871.
[19]	M. de León, J. C. Marrero and D. Martín de Diego, A new geometric setting for classical field theories, Banach Center Publ., 59 (2003), 189-209. doi: 10.4064/bc59-0-10.
[20]	M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory, North-Holland Math. Studies, 112, Elsevier Science Publishers B.V., Amsterdam, 1985.
[21]	M. de León and P. R. Rodrigues, Higher-order almost tangent geometry and non-autonomous Lagrangian dynamics, Rend. Circ. Mat. Palermo, 2 (1987), 157-171.
[22]	A. Echeverría-Enríquez, M. De León, M. C. Muñoz-Lecanda and N. Román-Roy, Extended Hamiltonian systems in multisymplectic field theories, J. Math. Phys., 48 (2007), 112901, 30pp. doi: 10.1063/1.2801875.
[23]	A. Echeverría-Enríquez, C. López, J. Marín-Solano, M. C. Muñoz-Lecanda and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for field theory, J. Math. Phys., 45 (2004), 360-380. doi: 10.1063/1.1628384.
[24]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector fields and connections: Setting Lagrangian equations in field theories, J. Math. Phys., 39 (1998), 4578-4603. doi: 10.1063/1.532525.
[25]	A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys., 41 (2000), 7402-7444. doi: 10.1063/1.1308075.
[26]	M. Ferraris and M. Francaviglia, Applications of the Poincaré-Cartan form in higher order field theories, in Differential Geometry and its Applications (Brno, 1986), Math. Appl. (East European Ser.) 27, Reidel, Dordrecht, 1987, 31-52.
[27]	M. Francaviglia and D. Krupka, The Hamiltonian formalism in higher order variational problems, Ann. Inst. H. Poincaré Sect. A (N.S.), 37 (1982), 295-315.
[28]	P. L. García and J. Muñoz-Masqué, On the geometrical structure of higher order variational calculus, in Procs. IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, Vol. I (Torino, 1982), Atti. Accad. Sci. Torino Cl. Sci. Fis. Math. Natur. 117, 1983, 127-147.
[29]	P. L. García and J. Muñoz-Masqué, Higher order regular variational problems, in Symplectic Geometry and Mathematical Physics (Aix-en-Provence, 1990), Progr. Math., 99, Birkhäuser Boston, MA, 1991, 136-159.
[30]	M. J. Gotay, A multisymplectic approach to the KdV equation, in Diferential Geometric Methods in Theoretical Physics (eds., K. Bleuler and M. Werner), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 250, Kluwer Acad. Publ., Dordrecht, 1988, 295-305.
[31]	K. Grabowska and L. Vitagliano, Tulczyjew triples in higher derivative field theory, J. Geom. Mech., 7 (2015), 1-33. doi: 10.3934/jgm.2015.7.1.
[32]	X. Gràcia, J. M. Pons and N. Román-Roy, Higher-order Lagrangian systems: Geometric structures, dynamics and constraints, J. Math. Phys., 32 (1991), 2744-2763. doi: 10.1063/1.529066.
[33]	X. Gràcia, J. M. Pons and N. Román-Roy, Higher-order conditions for singular Lagrangian systems, J. Phys. A: Math. Gen., 25 (1992), 1989-2004. doi: 10.1088/0305-4470/25/7/037.
[34]	D. R. Grigore, On a generalization of the Poincaré-Cartan form in higher-order field theory, in Variations, Geometry and Physics, Nova Sci. Publ., New York, 2009, 57-76.
[35]	M. Horák and I. Kolár, On the higher order Poincaré-Cartan forms, Czechoslovak Math. J., 33 (1983), 467-475.
[36]	I. Kolár, A geometrical version of the higher order Hamilton formalism in fibered manifolds, J. Geom. and Phys., 1 (1984), 127-137. doi: 10.1016/0393-0440(84)90007-X.
[37]	S. Kouranbaeva and S. Shkoller, A variational approach to second-order multisymplectic field theory, J. Geom. Phys., 35 (2000), 333-366. doi: 10.1016/S0393-0440(00)00012-7.
[38]	D. Krupka, On the higher order Hamilton theory in fibered spaces, in Procs. Conference on Differential Geometry and its Applications, Part 2, Univ. J. E. Purkyně, Brno, 1984, 167-183.
[39]	O. Krupkova, Higher-order mechanical systems with constraints, J. Math. Phys., 41 (2000), 5304-5324. doi: 10.1063/1.533411.
[40]	R. Miron, The Geometry of Higher-order Hamilton Spaces: Applications to Hamiltonian Mechanics, Fundamental Theories of Physics, 132, Springer-Verlag, New York. 2003. doi: 10.1007/978-94-010-0070-3.
[41]	P. Mukherjee and B. Paul, Gauge invariances of higher derivative Maxwell-Chern-Simons field theories: A new Hamiltonian approach, Phys. Rev. D, 85 (2012), 045028. doi: 10.1103/PhysRevD.85.045028.
[42]	J. Muñoz-Masqué, Canonical Cartan equations for higher order variational problems, J. Geom. Phys., 1 (1984), 1-7. doi: 10.1016/0393-0440(84)90001-9.
[43]	J. Muñoz-Masqué, Poincaré-Cartan forms in higher order variational calculus on fibred manifolds, Rev. Mat. Iberoamericana, 1 (1985), 85-126. doi: 10.4171/RMI/20.
[44]	P. D. Prieto-Martínez and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems, J. Phys. A Math. Theor., 44 (2011), 385203, 35pp. doi: 10.1088/1751-8113/44/38/385203.
[45]	P. D. Prieto-Martínez, N. Román-Roy, Unified formalism for higher-order non-autonomous dynamical systems, J. Math. Phys., 53 (2012), 032901, 38pp. doi: 10.1063/1.3692326.
[46]	P. D. Prieto-Martínez and N. Román-Roy, Higher-order mechanics: Variational principles and other topics, J. Geom. Mech., 5 (2013), 493-510. doi: 10.3934/jgm.2013.5.493.
[47]	A. M. Rey, N. Román-Roy and M. Salgado, Gunther's formalism k-symplectic formalism in classical field theory: Skinner-Rusk approach and the evolution operator, J. Math. Phys., 46 (2005), 052901, 24pp. doi: 10.1063/1.1876872.
[48]	A. M. Rey, N. Román-Roy, M. Salgado and S. Vilariño, k-cosymplectic classical field theories: Tulckzyjew and Skinner-Rusk formulations, Math. Phys. Anal. Geom., 15 (2012), 85-119. doi: 10.1007/s11040-012-9104-z.
[49]	D. J. Saunders, An alternative approach to the Cartan form in Lagrangian field theories, J. Phys. A, 20 (1987), 339-349. doi: 10.1088/0305-4470/20/2/019.
[50]	D. J. Saunders, The Geometry of Jet Bundles, London Mathematical Society, Lecture Notes Series, 142, Cambridge University Press, Cambridge, New York, 1989. doi: 10.1017/CBO9780511526411.
[51]	D. J. Saunders and M. Crampin, On the Legendre map in higher-order field theories, J. Phys. A: Math. Gen., 23 (1990), 3169-3182. doi: 10.1088/0305-4470/23/14/016.
[52]	R. Skinner and R. Rusk, Generalized Hamiltonian dynamics. I. Formulation on $T^Q\oplus TQ$, J. Math. Phys.,* 24 (1983), 2589-2594. doi: 10.1063/1.525654.
[53]	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.
[54]	P. F. Zhao and M. Z. Qin, Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation, J. Phys. A, 33 (2000), 3613-3626. doi: 10.1088/0305-4470/33/18/308.

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