A new multisymplectic unified formalism for second order classical field theories

Pedro Daniel  Prieto-Martínez; Narciso Román-Roy

doi:10.3934/jgm.2015.7.203

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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, (1978). Google Scholar
[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. doi: 10.1063/1.523904. Google Scholar
[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. doi: 10.1063/1.523905. Google Scholar
[4]	V. Aldaya and J. A. de Azcarraga, Higher order Hamiltonian formalism in field theory,, J. Phys. A, 13 (1980), 2545. doi: 10.1088/0305-4470/13/8/004. Google Scholar
[5]	U. M. Ascher and R. I. McLachlan, On symplectic and multisymplectic schemes for the KdV equation,, J. Sci. Comput., 25 (2005), 83. doi: 10.1007/s10915-004-4634-6. Google Scholar
[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. doi: 10.1088/1751-8113/40/40/005. Google Scholar
[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). doi: 10.1063/1.2929668. Google Scholar
[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. doi: 10.1088/0305-4470/21/12/013. Google Scholar
[9]	C. M. Campos, Geometric Methods in Classical Field Theory and Continous Media,, Ph.D. thesis, (2010). Google Scholar
[10]	C. M. Campos, Higher-order field theory with constraints,, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106* (2012), 89. doi: 10.1007/s13398-011-0025-7. Google Scholar*
[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). doi: 10.1088/1751-8113/42/47/475207. Google Scholar
[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. doi: 10.1017/S0305004100064501. Google Scholar
[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. doi: 10.1016/0926-2245(91)90013-Y. Google Scholar
[14]	L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3456158. Google Scholar
[15]	J. Cortés, S. Martínez and F. Cantrijn, Skinner-Rusk approach to time-dependent mechanics,, Physics Letters A, 300 (2002), 250. doi: 10.1016/S0375-9601(02)00777-6. Google Scholar
[16]	M. Crampin and D. J. Saunders, Homogeneity and projective equivalence of differential equation fields,, J. Geom. Mech., 4 (2012), 27. doi: 10.3934/jgm.2012.4.27. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. doi: 10.4064/bc59-0-10. Google Scholar
[20]	M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Math. Studies, (1985). Google Scholar
[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. Google Scholar
[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). doi: 10.1063/1.2801875. Google Scholar
[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. doi: 10.1063/1.1628384. Google Scholar
[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. doi: 10.1063/1.532525. Google Scholar
[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. doi: 10.1063/1.1308075. Google Scholar
[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), 31. Google Scholar
[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. Google Scholar
[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, (1982), 127. Google Scholar
[29]	P. L. García and J. Muñoz-Masqué, Higher order regular variational problems,, in Symplectic Geometry and Mathematical Physics* (Aix-en-Provence*, (1990), 136. Google Scholar
[30]	M. J. Gotay, A multisymplectic approach to the KdV equation,, in Diferential Geometric Methods in Theoretical Physics* (eds.*, (1988), 295. Google Scholar
[31]	K. Grabowska and L. Vitagliano, Tulczyjew triples in higher derivative field theory,, J. Geom. Mech., 7 (2015), 1. doi: 10.3934/jgm.2015.7.1. Google Scholar
[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. doi: 10.1063/1.529066. Google Scholar
[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. doi: 10.1088/0305-4470/25/7/037. Google Scholar
[34]	D. R. Grigore, On a generalization of the Poincaré-Cartan form in higher-order field theory,, in Variations, (2009), 57. Google Scholar
[35]	M. Horák and I. Kolár, On the higher order Poincaré-Cartan forms,, Czechoslovak Math. J., 33 (1983), 467. Google Scholar
[36]	I. Kolár, A geometrical version of the higher order Hamilton formalism in fibered manifolds,, J. Geom. and Phys., 1 (1984), 127. doi: 10.1016/0393-0440(84)90007-X. Google Scholar
[37]	S. Kouranbaeva and S. Shkoller, A variational approach to second-order multisymplectic field theory,, J. Geom. Phys., 35 (2000), 333. doi: 10.1016/S0393-0440(00)00012-7. Google Scholar
[38]	D. Krupka, On the higher order Hamilton theory in fibered spaces,, in Procs. Conference on Differential Geometry and its Applications, (1984), 167. Google Scholar
[39]	O. Krupkova, Higher-order mechanical systems with constraints,, J. Math. Phys., 41 (2000), 5304. doi: 10.1063/1.533411. Google Scholar
[40]	R. Miron, The Geometry of Higher-order Hamilton Spaces: Applications to Hamiltonian Mechanics,, Fundamental Theories of Physics, (2003). doi: 10.1007/978-94-010-0070-3. Google Scholar
[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). doi: 10.1103/PhysRevD.85.045028. Google Scholar
[42]	J. Muñoz-Masqué, Canonical Cartan equations for higher order variational problems,, J. Geom. Phys., 1 (1984), 1. doi: 10.1016/0393-0440(84)90001-9. Google Scholar
[43]	J. Muñoz-Masqué, Poincaré-Cartan forms in higher order variational calculus on fibred manifolds,, Rev. Mat. Iberoamericana, 1 (1985), 85. doi: 10.4171/RMI/20. Google Scholar
[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). doi: 10.1088/1751-8113/44/38/385203. Google Scholar
[45]	P. D. Prieto-Martínez, N. Román-Roy, Unified formalism for higher-order non-autonomous dynamical systems,, J. Math. Phys., 53 (2012). doi: 10.1063/1.3692326. Google Scholar
[46]	P. D. Prieto-Martínez and N. Román-Roy, Higher-order mechanics: Variational principles and other topics,, J. Geom. Mech., 5 (2013), 493. doi: 10.3934/jgm.2013.5.493. Google Scholar
[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). doi: 10.1063/1.1876872. Google Scholar
[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. doi: 10.1007/s11040-012-9104-z. Google Scholar
[49]	D. J. Saunders, An alternative approach to the Cartan form in Lagrangian field theories,, J. Phys. A, 20 (1987), 339. doi: 10.1088/0305-4470/20/2/019. Google Scholar
[50]	D. J. Saunders, The Geometry of Jet Bundles,, London Mathematical Society, (1989). doi: 10.1017/CBO9780511526411. Google Scholar
[51]	D. J. Saunders and M. Crampin, On the Legendre map in higher-order field theories,, J. Phys. A: Math. Gen., 23 (1990), 3169. doi: 10.1088/0305-4470/23/14/016. Google Scholar
[52]	R. Skinner and R. Rusk, Generalized Hamiltonian dynamics. I. Formulation on $T^Q\oplus TQ$,, J. Math. Phys., 24 (1983), 2589. doi: 10.1063/1.525654. Google Scholar*
[53]	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
[54]	P. F. Zhao and M. Z. Qin, Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation,, J. Phys. A, 33 (2000), 3613. doi: 10.1088/0305-4470/33/18/308. Google Scholar

show all references

References:

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics,, $2^{nd}$ edition, (1978). Google Scholar
[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. doi: 10.1063/1.523904. Google Scholar
[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. doi: 10.1063/1.523905. Google Scholar
[4]	V. Aldaya and J. A. de Azcarraga, Higher order Hamiltonian formalism in field theory,, J. Phys. A, 13 (1980), 2545. doi: 10.1088/0305-4470/13/8/004. Google Scholar
[5]	U. M. Ascher and R. I. McLachlan, On symplectic and multisymplectic schemes for the KdV equation,, J. Sci. Comput., 25 (2005), 83. doi: 10.1007/s10915-004-4634-6. Google Scholar
[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. doi: 10.1088/1751-8113/40/40/005. Google Scholar
[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). doi: 10.1063/1.2929668. Google Scholar
[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. doi: 10.1088/0305-4470/21/12/013. Google Scholar
[9]	C. M. Campos, Geometric Methods in Classical Field Theory and Continous Media,, Ph.D. thesis, (2010). Google Scholar
[10]	C. M. Campos, Higher-order field theory with constraints,, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106* (2012), 89. doi: 10.1007/s13398-011-0025-7. Google Scholar*
[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). doi: 10.1088/1751-8113/42/47/475207. Google Scholar
[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. doi: 10.1017/S0305004100064501. Google Scholar
[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. doi: 10.1016/0926-2245(91)90013-Y. Google Scholar
[14]	L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3456158. Google Scholar
[15]	J. Cortés, S. Martínez and F. Cantrijn, Skinner-Rusk approach to time-dependent mechanics,, Physics Letters A, 300 (2002), 250. doi: 10.1016/S0375-9601(02)00777-6. Google Scholar
[16]	M. Crampin and D. J. Saunders, Homogeneity and projective equivalence of differential equation fields,, J. Geom. Mech., 4 (2012), 27. doi: 10.3934/jgm.2012.4.27. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. doi: 10.4064/bc59-0-10. Google Scholar
[20]	M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Math. Studies, (1985). Google Scholar
[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. Google Scholar
[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). doi: 10.1063/1.2801875. Google Scholar
[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. doi: 10.1063/1.1628384. Google Scholar
[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. doi: 10.1063/1.532525. Google Scholar
[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. doi: 10.1063/1.1308075. Google Scholar
[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), 31. Google Scholar
[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. Google Scholar
[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, (1982), 127. Google Scholar
[29]	P. L. García and J. Muñoz-Masqué, Higher order regular variational problems,, in Symplectic Geometry and Mathematical Physics* (Aix-en-Provence*, (1990), 136. Google Scholar
[30]	M. J. Gotay, A multisymplectic approach to the KdV equation,, in Diferential Geometric Methods in Theoretical Physics* (eds.*, (1988), 295. Google Scholar
[31]	K. Grabowska and L. Vitagliano, Tulczyjew triples in higher derivative field theory,, J. Geom. Mech., 7 (2015), 1. doi: 10.3934/jgm.2015.7.1. Google Scholar
[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. doi: 10.1063/1.529066. Google Scholar
[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. doi: 10.1088/0305-4470/25/7/037. Google Scholar
[34]	D. R. Grigore, On a generalization of the Poincaré-Cartan form in higher-order field theory,, in Variations, (2009), 57. Google Scholar
[35]	M. Horák and I. Kolár, On the higher order Poincaré-Cartan forms,, Czechoslovak Math. J., 33 (1983), 467. Google Scholar
[36]	I. Kolár, A geometrical version of the higher order Hamilton formalism in fibered manifolds,, J. Geom. and Phys., 1 (1984), 127. doi: 10.1016/0393-0440(84)90007-X. Google Scholar
[37]	S. Kouranbaeva and S. Shkoller, A variational approach to second-order multisymplectic field theory,, J. Geom. Phys., 35 (2000), 333. doi: 10.1016/S0393-0440(00)00012-7. Google Scholar
[38]	D. Krupka, On the higher order Hamilton theory in fibered spaces,, in Procs. Conference on Differential Geometry and its Applications, (1984), 167. Google Scholar
[39]	O. Krupkova, Higher-order mechanical systems with constraints,, J. Math. Phys., 41 (2000), 5304. doi: 10.1063/1.533411. Google Scholar
[40]	R. Miron, The Geometry of Higher-order Hamilton Spaces: Applications to Hamiltonian Mechanics,, Fundamental Theories of Physics, (2003). doi: 10.1007/978-94-010-0070-3. Google Scholar
[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). doi: 10.1103/PhysRevD.85.045028. Google Scholar
[42]	J. Muñoz-Masqué, Canonical Cartan equations for higher order variational problems,, J. Geom. Phys., 1 (1984), 1. doi: 10.1016/0393-0440(84)90001-9. Google Scholar
[43]	J. Muñoz-Masqué, Poincaré-Cartan forms in higher order variational calculus on fibred manifolds,, Rev. Mat. Iberoamericana, 1 (1985), 85. doi: 10.4171/RMI/20. Google Scholar
[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). doi: 10.1088/1751-8113/44/38/385203. Google Scholar
[45]	P. D. Prieto-Martínez, N. Román-Roy, Unified formalism for higher-order non-autonomous dynamical systems,, J. Math. Phys., 53 (2012). doi: 10.1063/1.3692326. Google Scholar
[46]	P. D. Prieto-Martínez and N. Román-Roy, Higher-order mechanics: Variational principles and other topics,, J. Geom. Mech., 5 (2013), 493. doi: 10.3934/jgm.2013.5.493. Google Scholar
[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). doi: 10.1063/1.1876872. Google Scholar
[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. doi: 10.1007/s11040-012-9104-z. Google Scholar
[49]	D. J. Saunders, An alternative approach to the Cartan form in Lagrangian field theories,, J. Phys. A, 20 (1987), 339. doi: 10.1088/0305-4470/20/2/019. Google Scholar
[50]	D. J. Saunders, The Geometry of Jet Bundles,, London Mathematical Society, (1989). doi: 10.1017/CBO9780511526411. Google Scholar
[51]	D. J. Saunders and M. Crampin, On the Legendre map in higher-order field theories,, J. Phys. A: Math. Gen., 23 (1990), 3169. doi: 10.1088/0305-4470/23/14/016. Google Scholar
[52]	R. Skinner and R. Rusk, Generalized Hamiltonian dynamics. I. Formulation on $T^Q\oplus TQ$,, J. Math. Phys., 24 (1983), 2589. doi: 10.1063/1.525654. Google Scholar*
[53]	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
[54]	P. F. Zhao and M. Z. Qin, Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation,, J. Phys. A, 33 (2000), 3613. doi: 10.1088/0305-4470/33/18/308. Google Scholar

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American Institute of Mathematical Sciences

A new multisymplectic unified formalism for second order classical field theories

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A new multisymplectic unified formalism for second order classical field theories

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