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March  2020, 12(1): 85-106. doi: 10.3934/jgm.2020005

De Donder form for second order gravity

Jędrzej Śniatycki 1, and  Oǧul Esen 2,,

1. 

Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB. T2N 1N4, Canada
2. 

Department of Mathematics, Gebze Technical University, 41400 Gebze, Kocaeli, Turkey

* Corresponding author: Oǧul Esen

Received  March 2019 Published  January 2020

We show that the De Donder form for second order gravity, defined in terms of Ostrogradski's version of the Legendre transformation applied to all independent variables, is globally defined by its local coordinate descriptions. It is a natural differential operator applied to the diffeomorphism invariant Lagrangian of the theory.

Keywords: De Donder form, Second order gravity, Hilbert Lagrangian.
Mathematics Subject Classification: Primary: 83C05, 70H50; Secondary: 70S05.
Citation: Jędrzej Śniatycki, Oǧul Esen. De Donder form for second order gravity. Journal of Geometric Mechanics, 2020, 12 (1) : 85-106. doi: 10.3934/jgm.2020005
References:
[1]

V. Aldaya and J. A. de Azcárraga, Variational principles on rth. order jets of fibre bundles in field theory, Journal of Mathematical Physics, 19 (1978), 1876-1880.  doi: 10.1063/1.523904.

[2]

R. Arnowitt, S. Deser and C. W. Misner, The dynamics of general relativity", In Gravitation: An Introduction to Current Research, L. Witten (ed.), Wiley, New York, (1962), 227–265.

[3]

E. Binz, J. Śniatycki and H. Fischer, Geometry of Classical Fields, Elsevier Science Publishers, New York, 1988. Reprinted by Dover Publications, Mineola, N.Y., 2006.

[4]

C. M. Campos, Methods in Classical Field Theory and Continuous Media, Thesis, Departamento de Mathemáticas, Faculdad de Ciencias, Universidad Autónoma de Madrid, 2010.

[5]

S. Capriotti, Differential geometry, Palatini gravity and reduction, Journal of Mathematical Physics, 55 (2014), 012902, 29pp. doi: 10.1063/1.4862855.

[6]

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

[7]

É. Cartan, Les Espaces Métriques Fondés sur la Notion D'aire, Actualités Scientifique et Industrielles, no 72, 1933, Reprinted by Hermann, Paris, 1971.

[8]

M. Castrillón Lopez, J. Muñoz Masqué and E. Rosado María, First-order equivalent to Einstein-Hilbert Lagrangian, Journal of Mathematical Physics, 55 (2014), 082501, 9pp. doi: 10.1063/1.4890555.

[9]

Th. De Donder, Théorie invariantive du calcul des variations, Bull, Acad. de Belg., 1929, chap. 1; this reference appears in Cartan [7].

[10]

Th. De Donder, Théorie Invariantive du Calcul Des Variations, (nouvelle édit.), Gauthier Villars, Paris, 1935.

[11]

M. de Leon and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory, Elsevier Science Publishers, Amsterdam, 1985.

[12]

J. Gaset and N. Roman-Roy, Multisymplectic united formalism for Einstein-Hilbert gravity, Journal of Mathematical Physics, 59 (2018), 032502, 39pp. doi: 10.1063/1.4998526.

[13]

M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations I, In Mechanics, Analysis and Geometry, 200 Years after Lagrange, M. Fracaviglia (ed.), Elsevier Science Publishers, 1991,203–235.

[14]

I. V. Kanatchikov, From the De Donder-Weyl Hamiltonian formalism to quantization of gravity, In Current topics in Mathematical Cosmology, M. Rainer and H.J. Scmidt edts., World Scientific, Singapore, 1998,457–467.

[15]

Th. Lepage, Sur les champs géodé siques du calcul des variations, I et II, Académie Royale de Belgique, Bulletins de la Classe de Sciences, 5e série, 22 (1936), 716–729 and 1034–1046.

[16]

C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W.H. Freeman and Compant, 1973.

[17] P. Olver, Equivalence, Invariance and Symmetry, University Press, Cambridge, UK, 1995.  doi: 10.1017/CBO9780511609565.
[18]

M. Ostrogradski, Mémoires sur Les Équations Différentielles Relatives au Probléme des Isopé Rimètres, Mém. Ac. St. Petersburg, VI 4,385, 1850.

[19]

A. Palatini, Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton, Rend. Circ. Mat. Palermo, 43 (1919), 203-212. 

[20]

J. Śniatycki, On the canonical formulation of general relativity, Proceedings of Journees Relativistes, Faculté des Sciences, Caen, 1970,127–135.

[21]

J. Śniatycki, On the geometric structure of classical field theory in Lagrangian formulation, Proc. Camb. Phil. Soc., 68 (1970), 475-484.  doi: 10.1017/S0305004100046284.

[22]

J. Śniatycki and R. Segev, De Donder Construction for Higher Jets, arXiv: 1808: 03054v2, [math-phys], 2018.

[23]

W. Szczyrba, A symplectic structure in the set of Einstein metrics, Comm. Math. Phys., 80 (1976), 163-182. 

[24]

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

show all references

References:
[1]

V. Aldaya and J. A. de Azcárraga, Variational principles on rth. order jets of fibre bundles in field theory, Journal of Mathematical Physics, 19 (1978), 1876-1880.  doi: 10.1063/1.523904.

[2]

R. Arnowitt, S. Deser and C. W. Misner, The dynamics of general relativity", In Gravitation: An Introduction to Current Research, L. Witten (ed.), Wiley, New York, (1962), 227–265.

[3]

E. Binz, J. Śniatycki and H. Fischer, Geometry of Classical Fields, Elsevier Science Publishers, New York, 1988. Reprinted by Dover Publications, Mineola, N.Y., 2006.

[4]

C. M. Campos, Methods in Classical Field Theory and Continuous Media, Thesis, Departamento de Mathemáticas, Faculdad de Ciencias, Universidad Autónoma de Madrid, 2010.

[5]

S. Capriotti, Differential geometry, Palatini gravity and reduction, Journal of Mathematical Physics, 55 (2014), 012902, 29pp. doi: 10.1063/1.4862855.

[6]

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

[7]

É. Cartan, Les Espaces Métriques Fondés sur la Notion D'aire, Actualités Scientifique et Industrielles, no 72, 1933, Reprinted by Hermann, Paris, 1971.

[8]

M. Castrillón Lopez, J. Muñoz Masqué and E. Rosado María, First-order equivalent to Einstein-Hilbert Lagrangian, Journal of Mathematical Physics, 55 (2014), 082501, 9pp. doi: 10.1063/1.4890555.

[9]

Th. De Donder, Théorie invariantive du calcul des variations, Bull, Acad. de Belg., 1929, chap. 1; this reference appears in Cartan [7].

[10]

Th. De Donder, Théorie Invariantive du Calcul Des Variations, (nouvelle édit.), Gauthier Villars, Paris, 1935.

[11]

M. de Leon and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory, Elsevier Science Publishers, Amsterdam, 1985.

[12]

J. Gaset and N. Roman-Roy, Multisymplectic united formalism for Einstein-Hilbert gravity, Journal of Mathematical Physics, 59 (2018), 032502, 39pp. doi: 10.1063/1.4998526.

[13]

M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations I, In Mechanics, Analysis and Geometry, 200 Years after Lagrange, M. Fracaviglia (ed.), Elsevier Science Publishers, 1991,203–235.

[14]

I. V. Kanatchikov, From the De Donder-Weyl Hamiltonian formalism to quantization of gravity, In Current topics in Mathematical Cosmology, M. Rainer and H.J. Scmidt edts., World Scientific, Singapore, 1998,457–467.

[15]

Th. Lepage, Sur les champs géodé siques du calcul des variations, I et II, Académie Royale de Belgique, Bulletins de la Classe de Sciences, 5e série, 22 (1936), 716–729 and 1034–1046.

[16]

C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W.H. Freeman and Compant, 1973.

[17] P. Olver, Equivalence, Invariance and Symmetry, University Press, Cambridge, UK, 1995.  doi: 10.1017/CBO9780511609565.
[18]

M. Ostrogradski, Mémoires sur Les Équations Différentielles Relatives au Probléme des Isopé Rimètres, Mém. Ac. St. Petersburg, VI 4,385, 1850.

[19]

A. Palatini, Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton, Rend. Circ. Mat. Palermo, 43 (1919), 203-212. 

[20]

J. Śniatycki, On the canonical formulation of general relativity, Proceedings of Journees Relativistes, Faculté des Sciences, Caen, 1970,127–135.

[21]

J. Śniatycki, On the geometric structure of classical field theory in Lagrangian formulation, Proc. Camb. Phil. Soc., 68 (1970), 475-484.  doi: 10.1017/S0305004100046284.

[22]

J. Śniatycki and R. Segev, De Donder Construction for Higher Jets, arXiv: 1808: 03054v2, [math-phys], 2018.

[23]

W. Szczyrba, A symplectic structure in the set of Einstein metrics, Comm. Math. Phys., 80 (1976), 163-182. 

[24]

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

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