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September  2019, 11(3): 361-396. doi: 10.3934/jgm.2019019

New multisymplectic approach to the Metric-Affine (Einstein-Palatini) action for gravity

Jordi Gaset and  Narciso Román-Roy

Department of Mathematics, Ed. C-3, Campus Norte UPC, C/ Jordi Girona 1. 08034 Barcelona, Spain

Received  June 2018 Revised  March 2019 Published  August 2019

We present a covariant multisymplectic formulation for the Einstein-Palatini (or Metric-Affine) model of General Relativity (without energy-matter sources). As it is described by a first-order affine Lagrangian (in the derivatives of the fields), it is singular and, hence, this is a gauge field theory with constraints. These constraints are obtained after applying a constraint algorithm to the field equations, both in the Lagrangian and the Hamiltonian formalisms. In order to do this, the covariant field equations must be written in a suitable geometrical way, using integrable distributions which are represented by multivector fields of a certain type. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomentum (covariant) Hamiltonian formalism. The gauge symmetries of the model are discussed in both formalisms and, from them, the equivalence with the Einstein-Hilbert model is established.

Keywords: Classical field theories, jet bundles, multisymplectic forms, Einstein-Palatini action, metric-affine models, constraints, gauge symmetries.
Mathematics Subject Classification: Primary: 53D42, 55R10, 70S05, 83C05; Secondary: 49S05, 53C15, 53C80, 53Z05.
Citation: Jordi Gaset, Narciso Román-Roy. New multisymplectic approach to the Metric-Affine (Einstein-Palatini) action for gravity. Journal of Geometric Mechanics, 2019, 11 (3) : 361-396. doi: 10.3934/jgm.2019019
References:
[1]

V. Aldaya and J. A. de Azcárraga, Geometric formulation of classical mechanics and field theory, Riv. Nuovo Cimento, 3 (1980), 1-66.  doi: 10.1007/BF02906204.  Google Scholar

[2]

M. J. Bergvelt and E. A. de Kerf, The Hamiltonian structure of Yang-Mills theories and instantons (Part Ⅰ), Physica, 139A (1986), 101-124.  doi: 10.1016/0378-4371(86)90007-5.  Google Scholar

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J. Berra-Montiel, A. Molgado and D. Serrano-Blanco, De Donder-Weyl Hamiltonian formalism of MacDowell-Mansouri gravity, Class. Quant. Grav., 34 (2017), 235002, 14pp.  doi: 10.1088/1361-6382/aa924a.  Google Scholar

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S. Capriotti, Differential geometry, Palatini gravity and reduction, J. Math. Phys., 55 (2014), 012902, 29pp.  doi: 10.1063/1.4862855.  Google Scholar

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S. Capriotti, Unified formalism for Palatini gravity, Int. J. Geom. Meth. Mod. Phys., 15 (2018), 1850044, 33pp.  doi: 10.1142/S0219887818500445.  Google Scholar

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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.  Google Scholar

[7]

M. Castrillón, J. Muñoz-Masqué and M. E. Rosado, First-order equivalent to Einstein-Hilbert Lagrangian, J. Math. Phys., 55 (2014), 082501, 9pp.  doi: 10.1063/1.4890555.  Google Scholar

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R. Cianci, S. Vignolo and D. Bruno, General Relativity as a constrained gauge theory, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 1493-1500.  doi: 10.1142/S0219887806001818.  Google Scholar

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C. Cremaschini and M. Tessarotto, Manifest covariant Hamiltonian theory of general relativity, App. Phys. Research, 8 (2016), 60-81.  doi: 10.5539/apr.v8n2p60.  Google Scholar

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C. Cremaschini and M. Tessarotto, Hamiltonian approach to GR-Part 1: Covariant theory of classical gravity, Eur. Phys. Journal C, 77 (2017), 329.  doi: 10.1140/epjc/s10052-017-4854-1.  Google Scholar

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N. Dadhich and J. M. Pons, On the equivalence of the Einstein–Hilbert and the Einstein–Palatini formulations of general relativity for an arbitrary connection, Gen. Rel. Grav., 44 (2012), 2337-2352.  doi: 10.1007/s10714-012-1393-9.  Google Scholar

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M. de León, J. Marín-Solano and J. C. Marrero, A geometrical approach to classical field theories: A constraint algorithm for singular theories, in New Developments in Differential Geometry, (Debrecen, 1994, eds L. Tamassi and J. Szenthe), Math. Appl. 350, Kluwer Acad. Publ., Dordrecht, 1996, 291–312. doi: 10.1007/978-94-009-0149-0_22.  Google Scholar

[13]

M. de León, J. Marín–Solano, J. C. Marrero, M. C. Muñoz–Lecanda and N. Román-Roy, Singular Lagrangian systems on jet bundles, Fortsch. Phys., 50 (2002), 105-169.  doi: 10.1002/1521-3978(200203)50:2<105::AID-PROP105>3.0.CO;2-N.  Google Scholar

[14]

M. de León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz–Lecanda and N. Román-Roy, Pre-multisymplectic constraint algorithm for field theories, Int. J. Geom. Meth. Mod. Phys., 2 (2005), 839-871.  doi: 10.1142/S0219887805000880.  Google Scholar

[15]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometry of Lagrangian first-order classical field theories, Forts. Phys., 44 (1996), 235-280.  doi: 10.1002/prop.2190440304.  Google Scholar

[16]

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.  Google Scholar

[17]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of hamiltonian field theories: equations and symmetries, J. Phys. A: Math. Gen., 32 (1999), 8461-8484.  doi: 10.1088/0305-4470/32/48/309.  Google Scholar

[18]

A. Echeverría–Enríquez, M. C. Muñoz–Lecanda and N. Román-Roy, On the multimomentum bundles and the Legendre maps in field theories, Rep. Math. Phys., 45 (2000), 85-105.  doi: 10.1016/S0034-4877(00)88873-4.  Google Scholar

[19]

A. Echeverría–Enríquez, M. C. Muñoz–Lecanda and N. Román–Roy, Connections and jet fields, preprint, arXiv: 1803.10451[math.DG] (2018). Google Scholar

[20]

A. Einstein, Einheitliche Fieldtheorie von Gravitation und Elektrizit t, Pruess. Akad.Wiss., 414, (1925); A. Unzicker and T. Case, Translation of Einstein's attempt of a unified field theory with teleparallelism, preprint, arXiv: physics/0503046[11]. Google Scholar

[21]

G. Esposito, C. Stornaiolo and G. Gionti, Spacetime covariant form of Ashtekar's constraints, Nuovo Cim.B, 110 (1995), 1137-1152.  doi: 10.1007/BF02724605.  Google Scholar

[22]

P. L. García, The Poincaré–Cartan invariant in the calculus of variations, Symp. Math., (1974), 219-246.   Google Scholar

[23]

J. Gaset, P. D. Prieto–Martínez and N. Román–Roy, Variational principles and symmetries on fibered multisymplectic manifolds, Comm. in Maths., 24 (2016), 137-152.  doi: 10.1515/cm-2016-0010.  Google Scholar

[24]

J. Gaset and N. Román–Roy, Order reduction, projectability and constraints of second–order field theories and higher-order mechanics, Rep. Math. Phys., 78 (2016), 327–337. https://doi.org/10.1063/1.4940047. doi: 10.1016/S0034-4877(17)30012-5.  Google Scholar

[25]

J. Gaset and N. Román-Roy, Multisymplectic unified formalism for Einstein-Hilbert gravity, J. Math. Phys., 59 (2018), 032502, 39pp.  doi: 10.1063/1.4998526.  Google Scholar

[26]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, New Lagrangian and Hamiltonian Methods in Field Theory, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. (ISBN: 981-02-1587-8.). doi: 10.1142/2199.  Google Scholar

[27]

H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier Grenoble, 23 (1973), 203-267.  doi: 10.5802/aif.451.  Google Scholar

[28]

M. J. Gotay, J. Isenberg, J. E. Marsden and R. Montgomery, Momentum maps and classical relativistic fields. I. Covariant theory, preprint, arXiv: physics/9801019[math-ph] (2004). Google Scholar

[29]

M. J. Gotay and J.M. Nester, Presymplectic Hamilton and Lagrange systems, gauge transformations and the Dirac theory of constraints, in Group Theoretical Methods in Physics (eds. W. Beigelbock, A. Böhm, E. Takasugi), Lect. Notes in Phys., Springer, Berlin, 94 (1979), 272–279. doi: 10.1007/3-540-09238-2_74.  Google Scholar

[30]

X. Grácia, J. M. Pons and N. Román-Roy, Higher order conditions for singular Lagrangian dynamics, J. Phys. A: Math. Gen., 25 (1992), 1989-2004.  doi: 10.1088/0305-4470/25/7/037.  Google Scholar

[31]

A. Ibort and A. Spivak, On a covariant Hamiltonian description of Palatini's gravity on manifolds with boundary, preprint, arXiv: 1605.03492[math-ph] (2016). Google Scholar

[32]

I. V. Kanatchikov, Precanonical quantum gravity: Quantization without the space-time decomposition, Int. J. Theor. Phys., 40 (2001), 1121-1149.  doi: 10.1023/A:1017557603606.  Google Scholar

[33]

I. V. Kanatchikov, On precanonical quantization of gravity, Nonlin. Phenom. Complex Sys., (NPCS) 17 (2014), 372–376.  Google Scholar

[34]

I. V. Kanatchikov, On the `spin connection foam' picture of quantum gravity from precanonical quantization, in Procs. 14th Marcel Grossmann Meeting on General Relativity: "Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories", U. Rome "La Sapienza", Italy 2015, (2017), 3907–3915. doi: 10.1142/9789813226609_0519.  Google Scholar

[35] D. Krupka, Introduction to Global Variational Geometry, Atlantis Studies in Variational Geometry, Atlantis Press, 2015.   Google Scholar
[36]

D. Krupka and O. Stepankova, On the Hamilton form in second order calculus of variations, in Procs. Int. Meeting on Geometry and Physics, Florence 1982, Pitagora, Bologna, 1983, 85–101.  Google Scholar

[37]

M. Montesinos, D. González, M. Celad and B. Díaz, Reformulation of the symmetries of first-order general relativity, Class. Quant. Grav., 34 (2017), 205002, 13pp.  doi: 10.1088/1361-6382/aa89f3.  Google Scholar

[38]

J. Muñoz-Masqué and M. E. Rosado, Diffeomorphism-invariant covariant Hamiltonians of a pseudo-Riemannian metric and a linear connection, Adv. Theor. Math. Phys., 16 (2012), 851-886.  doi: 10.4310/ATMP.2012.v16.n3.a3.  Google Scholar

[39]

P. D. Prieto Martínez and N. Román-Roy, A new multisymplectic unified formalism for second-order classical field theories, J. Geom. Mech., 7 (2015), 203-253.  doi: 10.3934/jgm.2015.7.203.  Google Scholar

[40]

N. Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories, Symm. Integ. Geom. Methods Appl. (SIGMA), 5 (2009), Paper 100, 25 pp. doi: 10.3842/SIGMA.2009.100.  Google Scholar

[41]

M. E. Rosado and J. Muñoz-Masqué, Integrability of second-order Lagrangians admitting a first-order Hamiltonian formalism, Diff. Geom. Apps., 35 (2014), 164-177.  doi: 10.1016/j.difgeo.2014.04.006.  Google Scholar

[42]

M. E. Rosado and J. Muñoz-Masqué, Second-order Lagrangians admitting a first-order Hamiltonian formalism, J. Annali di Matematica, 197 (2018), 357-397.  doi: 10.1007/s10231-017-0683-y.  Google Scholar

[43]

C. Rovelli, A note on the foundation of relativistic mechanics. Ⅱ: Covariant Hamiltonian general relativity, in Topics in Mathematical Physics, General Relativity and Cosmology (eds. H. Garcia-Compean, B. Mielnik, M. Montesinos, M. Przanowski), 397–407, World Sci. Publ., Hackensack, NJ, 2006. doi: 10.1142/9789812772732_0033.  Google Scholar

[44]

G. Sardanashvily, Generalized Hamiltonian Formalism for Field Theory. Constraint Systems, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. doi: 10.1142/9789812831484.  Google Scholar

[45]

D. J. Saunders, The Geometry of Jet Bundles, London Mathematical Society, Lecture notes series, 142, Cambridge University Press, Cambridge, New York 1989. (ISBN-13: 978-0521369480). doi: 10.1017/CBO9780511526411.  Google Scholar

[46]

C. G. Torre, Local cohomology in field theory (with applications to the Einstein equations), preprint, arXiv: hep-th/9706092 (1997). Google Scholar

[47]

D. Vey, Multisymplectic formulation of vielbein gravity. De Donder-Weyl formulation, Hamiltonian (n-1)-forms, Class. Quantum Grav., 32 (2015), 095005, 50pp. doi: 10.1088/0264-9381/32/9/095005.  Google Scholar

show all references

References:
[1]

V. Aldaya and J. A. de Azcárraga, Geometric formulation of classical mechanics and field theory, Riv. Nuovo Cimento, 3 (1980), 1-66.  doi: 10.1007/BF02906204.  Google Scholar

[2]

M. J. Bergvelt and E. A. de Kerf, The Hamiltonian structure of Yang-Mills theories and instantons (Part Ⅰ), Physica, 139A (1986), 101-124.  doi: 10.1016/0378-4371(86)90007-5.  Google Scholar

[3]

J. Berra-Montiel, A. Molgado and D. Serrano-Blanco, De Donder-Weyl Hamiltonian formalism of MacDowell-Mansouri gravity, Class. Quant. Grav., 34 (2017), 235002, 14pp.  doi: 10.1088/1361-6382/aa924a.  Google Scholar

[4]

S. Capriotti, Differential geometry, Palatini gravity and reduction, J. Math. Phys., 55 (2014), 012902, 29pp.  doi: 10.1063/1.4862855.  Google Scholar

[5]

S. Capriotti, Unified formalism for Palatini gravity, Int. J. Geom. Meth. Mod. Phys., 15 (2018), 1850044, 33pp.  doi: 10.1142/S0219887818500445.  Google Scholar

[6]

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.  Google Scholar

[7]

M. Castrillón, J. Muñoz-Masqué and M. E. Rosado, First-order equivalent to Einstein-Hilbert Lagrangian, J. Math. Phys., 55 (2014), 082501, 9pp.  doi: 10.1063/1.4890555.  Google Scholar

[8]

R. Cianci, S. Vignolo and D. Bruno, General Relativity as a constrained gauge theory, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 1493-1500.  doi: 10.1142/S0219887806001818.  Google Scholar

[9]

C. Cremaschini and M. Tessarotto, Manifest covariant Hamiltonian theory of general relativity, App. Phys. Research, 8 (2016), 60-81.  doi: 10.5539/apr.v8n2p60.  Google Scholar

[10]

C. Cremaschini and M. Tessarotto, Hamiltonian approach to GR-Part 1: Covariant theory of classical gravity, Eur. Phys. Journal C, 77 (2017), 329.  doi: 10.1140/epjc/s10052-017-4854-1.  Google Scholar

[11]

N. Dadhich and J. M. Pons, On the equivalence of the Einstein–Hilbert and the Einstein–Palatini formulations of general relativity for an arbitrary connection, Gen. Rel. Grav., 44 (2012), 2337-2352.  doi: 10.1007/s10714-012-1393-9.  Google Scholar

[12]

M. de León, J. Marín-Solano and J. C. Marrero, A geometrical approach to classical field theories: A constraint algorithm for singular theories, in New Developments in Differential Geometry, (Debrecen, 1994, eds L. Tamassi and J. Szenthe), Math. Appl. 350, Kluwer Acad. Publ., Dordrecht, 1996, 291–312. doi: 10.1007/978-94-009-0149-0_22.  Google Scholar

[13]

M. de León, J. Marín–Solano, J. C. Marrero, M. C. Muñoz–Lecanda and N. Román-Roy, Singular Lagrangian systems on jet bundles, Fortsch. Phys., 50 (2002), 105-169.  doi: 10.1002/1521-3978(200203)50:2<105::AID-PROP105>3.0.CO;2-N.  Google Scholar

[14]

M. de León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz–Lecanda and N. Román-Roy, Pre-multisymplectic constraint algorithm for field theories, Int. J. Geom. Meth. Mod. Phys., 2 (2005), 839-871.  doi: 10.1142/S0219887805000880.  Google Scholar

[15]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometry of Lagrangian first-order classical field theories, Forts. Phys., 44 (1996), 235-280.  doi: 10.1002/prop.2190440304.  Google Scholar

[16]

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.  Google Scholar

[17]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of hamiltonian field theories: equations and symmetries, J. Phys. A: Math. Gen., 32 (1999), 8461-8484.  doi: 10.1088/0305-4470/32/48/309.  Google Scholar

[18]

A. Echeverría–Enríquez, M. C. Muñoz–Lecanda and N. Román-Roy, On the multimomentum bundles and the Legendre maps in field theories, Rep. Math. Phys., 45 (2000), 85-105.  doi: 10.1016/S0034-4877(00)88873-4.  Google Scholar

[19]

A. Echeverría–Enríquez, M. C. Muñoz–Lecanda and N. Román–Roy, Connections and jet fields, preprint, arXiv: 1803.10451[math.DG] (2018). Google Scholar

[20]

A. Einstein, Einheitliche Fieldtheorie von Gravitation und Elektrizit t, Pruess. Akad.Wiss., 414, (1925); A. Unzicker and T. Case, Translation of Einstein's attempt of a unified field theory with teleparallelism, preprint, arXiv: physics/0503046[11]. Google Scholar

[21]

G. Esposito, C. Stornaiolo and G. Gionti, Spacetime covariant form of Ashtekar's constraints, Nuovo Cim.B, 110 (1995), 1137-1152.  doi: 10.1007/BF02724605.  Google Scholar

[22]

P. L. García, The Poincaré–Cartan invariant in the calculus of variations, Symp. Math., (1974), 219-246.   Google Scholar

[23]

J. Gaset, P. D. Prieto–Martínez and N. Román–Roy, Variational principles and symmetries on fibered multisymplectic manifolds, Comm. in Maths., 24 (2016), 137-152.  doi: 10.1515/cm-2016-0010.  Google Scholar

[24]

J. Gaset and N. Román–Roy, Order reduction, projectability and constraints of second–order field theories and higher-order mechanics, Rep. Math. Phys., 78 (2016), 327–337. https://doi.org/10.1063/1.4940047. doi: 10.1016/S0034-4877(17)30012-5.  Google Scholar

[25]

J. Gaset and N. Román-Roy, Multisymplectic unified formalism for Einstein-Hilbert gravity, J. Math. Phys., 59 (2018), 032502, 39pp.  doi: 10.1063/1.4998526.  Google Scholar

[26]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, New Lagrangian and Hamiltonian Methods in Field Theory, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. (ISBN: 981-02-1587-8.). doi: 10.1142/2199.  Google Scholar

[27]

H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier Grenoble, 23 (1973), 203-267.  doi: 10.5802/aif.451.  Google Scholar

[28]

M. J. Gotay, J. Isenberg, J. E. Marsden and R. Montgomery, Momentum maps and classical relativistic fields. I. Covariant theory, preprint, arXiv: physics/9801019[math-ph] (2004). Google Scholar

[29]

M. J. Gotay and J.M. Nester, Presymplectic Hamilton and Lagrange systems, gauge transformations and the Dirac theory of constraints, in Group Theoretical Methods in Physics (eds. W. Beigelbock, A. Böhm, E. Takasugi), Lect. Notes in Phys., Springer, Berlin, 94 (1979), 272–279. doi: 10.1007/3-540-09238-2_74.  Google Scholar

[30]

X. Grácia, J. M. Pons and N. Román-Roy, Higher order conditions for singular Lagrangian dynamics, J. Phys. A: Math. Gen., 25 (1992), 1989-2004.  doi: 10.1088/0305-4470/25/7/037.  Google Scholar

[31]

A. Ibort and A. Spivak, On a covariant Hamiltonian description of Palatini's gravity on manifolds with boundary, preprint, arXiv: 1605.03492[math-ph] (2016). Google Scholar

[32]

I. V. Kanatchikov, Precanonical quantum gravity: Quantization without the space-time decomposition, Int. J. Theor. Phys., 40 (2001), 1121-1149.  doi: 10.1023/A:1017557603606.  Google Scholar

[33]

I. V. Kanatchikov, On precanonical quantization of gravity, Nonlin. Phenom. Complex Sys., (NPCS) 17 (2014), 372–376.  Google Scholar

[34]

I. V. Kanatchikov, On the `spin connection foam' picture of quantum gravity from precanonical quantization, in Procs. 14th Marcel Grossmann Meeting on General Relativity: "Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories", U. Rome "La Sapienza", Italy 2015, (2017), 3907–3915. doi: 10.1142/9789813226609_0519.  Google Scholar

[35] D. Krupka, Introduction to Global Variational Geometry, Atlantis Studies in Variational Geometry, Atlantis Press, 2015.   Google Scholar
[36]

D. Krupka and O. Stepankova, On the Hamilton form in second order calculus of variations, in Procs. Int. Meeting on Geometry and Physics, Florence 1982, Pitagora, Bologna, 1983, 85–101.  Google Scholar

[37]

M. Montesinos, D. González, M. Celad and B. Díaz, Reformulation of the symmetries of first-order general relativity, Class. Quant. Grav., 34 (2017), 205002, 13pp.  doi: 10.1088/1361-6382/aa89f3.  Google Scholar

[38]

J. Muñoz-Masqué and M. E. Rosado, Diffeomorphism-invariant covariant Hamiltonians of a pseudo-Riemannian metric and a linear connection, Adv. Theor. Math. Phys., 16 (2012), 851-886.  doi: 10.4310/ATMP.2012.v16.n3.a3.  Google Scholar

[39]

P. D. Prieto Martínez and N. Román-Roy, A new multisymplectic unified formalism for second-order classical field theories, J. Geom. Mech., 7 (2015), 203-253.  doi: 10.3934/jgm.2015.7.203.  Google Scholar

[40]

N. Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories, Symm. Integ. Geom. Methods Appl. (SIGMA), 5 (2009), Paper 100, 25 pp. doi: 10.3842/SIGMA.2009.100.  Google Scholar

[41]

M. E. Rosado and J. Muñoz-Masqué, Integrability of second-order Lagrangians admitting a first-order Hamiltonian formalism, Diff. Geom. Apps., 35 (2014), 164-177.  doi: 10.1016/j.difgeo.2014.04.006.  Google Scholar

[42]

M. E. Rosado and J. Muñoz-Masqué, Second-order Lagrangians admitting a first-order Hamiltonian formalism, J. Annali di Matematica, 197 (2018), 357-397.  doi: 10.1007/s10231-017-0683-y.  Google Scholar

[43]

C. Rovelli, A note on the foundation of relativistic mechanics. Ⅱ: Covariant Hamiltonian general relativity, in Topics in Mathematical Physics, General Relativity and Cosmology (eds. H. Garcia-Compean, B. Mielnik, M. Montesinos, M. Przanowski), 397–407, World Sci. Publ., Hackensack, NJ, 2006. doi: 10.1142/9789812772732_0033.  Google Scholar

[44]

G. Sardanashvily, Generalized Hamiltonian Formalism for Field Theory. Constraint Systems, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. doi: 10.1142/9789812831484.  Google Scholar

[45]

D. J. Saunders, The Geometry of Jet Bundles, London Mathematical Society, Lecture notes series, 142, Cambridge University Press, Cambridge, New York 1989. (ISBN-13: 978-0521369480). doi: 10.1017/CBO9780511526411.  Google Scholar

[46]

C. G. Torre, Local cohomology in field theory (with applications to the Einstein equations), preprint, arXiv: hep-th/9706092 (1997). Google Scholar

[47]

D. Vey, Multisymplectic formulation of vielbein gravity. De Donder-Weyl formulation, Hamiltonian (n-1)-forms, Class. Quantum Grav., 32 (2015), 095005, 50pp. doi: 10.1088/0264-9381/32/9/095005.  Google Scholar

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