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

De Donder form for second order gravity

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.

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[16]

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

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

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[16]

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

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

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

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