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

A new multisymplectic unified formalism for second order classical field theories

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

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.
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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