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.

[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.

[1]

Nicolás Borda, Javier Fernández, Sergio Grillo. Discrete second order constrained Lagrangian systems: First results. Journal of Geometric Mechanics, 2013, 5 (4) : 381-397. doi: 10.3934/jgm.2013.5.381

[2]

Jun-Ren Luo, Ti-Jun Xiao. Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping. Evolution Equations and Control Theory, 2020, 9 (2) : 359-373. doi: 10.3934/eect.2020009

[3]

Kun-Peng Jin, Jin Liang, Ti-Jun Xiao. Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3141-3166. doi: 10.3934/dcdss.2021077

[4]

Elisabetta Carlini, Francisco J. Silva. A semi-Lagrangian scheme for a degenerate second order mean field game system. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4269-4292. doi: 10.3934/dcds.2015.35.4269

[5]

Jean Mawhin. Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1065-1076. doi: 10.3934/dcdss.2013.6.1065

[6]

Kaizhi Wang, Yong Li. Existence and monotonicity property of minimizers of a nonconvex variational problem with a second-order Lagrangian. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 687-699. doi: 10.3934/dcds.2009.25.687

[7]

Sandro Zagatti. Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 2005-2025. doi: 10.3934/dcds.2021181

[8]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643

[9]

B. Bonnard, J.-B. Caillau, E. Trélat. Second order optimality conditions with applications. Conference Publications, 2007, 2007 (Special) : 145-154. doi: 10.3934/proc.2007.2007.145

[10]

Carla Mascia, Giancarlo Rinaldo, Massimiliano Sala. Hilbert quasi-polynomial for order domains and application to coding theory. Advances in Mathematics of Communications, 2018, 12 (2) : 287-301. doi: 10.3934/amc.2018018

[11]

Karthikeyan Rajagopal, Serdar Cicek, Akif Akgul, Sajad Jafari, Anitha Karthikeyan. Chaotic cuttlesh: king of camouage with self-excited and hidden flows, its fractional-order form and communication designs with fractional form. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1001-1013. doi: 10.3934/dcdsb.2019205

[12]

Cédric M. Campos, Elisa Guzmán, Juan Carlos Marrero. Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (1) : 1-26. doi: 10.3934/jgm.2012.4.1

[13]

Antonio Marigonda. Second order conditions for the controllability of nonlinear systems with drift. Communications on Pure and Applied Analysis, 2006, 5 (4) : 861-885. doi: 10.3934/cpaa.2006.5.861

[14]

Fausto Ferrari. Mean value properties of fractional second order operators. Communications on Pure and Applied Analysis, 2015, 14 (1) : 83-106. doi: 10.3934/cpaa.2015.14.83

[15]

François Hamel, Emmanuel Russ, Nikolai Nadirashvili. Comparisons of eigenvalues of second order elliptic operators. Conference Publications, 2007, 2007 (Special) : 477-486. doi: 10.3934/proc.2007.2007.477

[16]

Bi Ping, Maoan Han. Oscillation of second order difference equations with advanced argument. Conference Publications, 2003, 2003 (Special) : 108-112. doi: 10.3934/proc.2003.2003.108

[17]

Jiann-Sheng Jiang, Kung-Hwang Kuo, Chi-Kun Lin. Homogenization of second order equation with spatial dependent coefficient. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 303-313. doi: 10.3934/dcds.2005.12.303

[18]

José F. Cariñena, Javier de Lucas Araujo. Superposition rules and second-order Riccati equations. Journal of Geometric Mechanics, 2011, 3 (1) : 1-22. doi: 10.3934/jgm.2011.3.1

[19]

Ahmed Y. Abdallah. Exponential attractors for second order lattice dynamical systems. Communications on Pure and Applied Analysis, 2009, 8 (3) : 803-813. doi: 10.3934/cpaa.2009.8.803

[20]

Norimichi Hirano, Zhi-Qiang Wang. Subharmonic solutions for second order Hamiltonian systems. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 467-474. doi: 10.3934/dcds.1998.4.467

2020 Impact Factor: 0.857

Metrics

  • PDF downloads (218)
  • HTML views (206)
  • Cited by (0)

Other articles
by authors

[Back to Top]