-
Previous Article
Self-organization on Riemannian manifolds
- JGM Home
- This Issue
-
Next Article
A symmetry-adapted numerical scheme for SDEs
New multisymplectic approach to the Metric-Affine (Einstein-Palatini) action for gravity
Department of Mathematics, Ed. C-3, Campus Norte UPC, C/ Jordi Girona 1. 08034 Barcelona, Spain |
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.
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. |
[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. |
[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. |
[4] |
S. Capriotti,
Differential geometry, Palatini gravity and reduction, J. Math. Phys., 55 (2014), 012902, 29pp.
doi: 10.1063/1.4862855. |
[5] |
S. Capriotti,
Unified formalism for Palatini gravity, Int. J. Geom. Meth. Mod. Phys., 15 (2018), 1850044, 33pp.
doi: 10.1142/S0219887818500445. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
P. L. García,
The Poincaré–Cartan invariant in the calculus of variations, Symp. Math., (1974), 219-246.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[33] |
I. V. Kanatchikov, On precanonical quantization of gravity, Nonlin. Phenom. Complex Sys., (NPCS) 17 (2014), 372–376. |
[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. |
[35] |
D. Krupka, Introduction to Global Variational Geometry, Atlantis Studies in Variational Geometry, Atlantis Press, 2015.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[44] |
G. Sardanashvily, Generalized Hamiltonian Formalism for Field Theory. Constraint Systems, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
doi: 10.1142/9789812831484. |
[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. |
[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. |
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. |
[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. |
[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. |
[4] |
S. Capriotti,
Differential geometry, Palatini gravity and reduction, J. Math. Phys., 55 (2014), 012902, 29pp.
doi: 10.1063/1.4862855. |
[5] |
S. Capriotti,
Unified formalism for Palatini gravity, Int. J. Geom. Meth. Mod. Phys., 15 (2018), 1850044, 33pp.
doi: 10.1142/S0219887818500445. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
P. L. García,
The Poincaré–Cartan invariant in the calculus of variations, Symp. Math., (1974), 219-246.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[33] |
I. V. Kanatchikov, On precanonical quantization of gravity, Nonlin. Phenom. Complex Sys., (NPCS) 17 (2014), 372–376. |
[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. |
[35] |
D. Krupka, Introduction to Global Variational Geometry, Atlantis Studies in Variational Geometry, Atlantis Press, 2015.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[44] |
G. Sardanashvily, Generalized Hamiltonian Formalism for Field Theory. Constraint Systems, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
doi: 10.1142/9789812831484. |
[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. |
[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. |
[1] |
Alvaro Sandroni, Eran Shmaya. A prequential test for exchangeable theories. Journal of Dynamics & Games, 2014, 1 (3) : 497-505. doi: 10.3934/jdg.2014.1.497 |
[2] |
David Cantala, Juan Sebastián Pereyra. Endogenous budget constraints in the assignment game. Journal of Dynamics & Games, 2015, 2 (3&4) : 207-225. doi: 10.3934/jdg.2015002 |
[3] |
Gelasio Salaza, Edgardo Ugalde, Jesús Urías. Master--slave synchronization of affine cellular automaton pairs. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 491-502. doi: 10.3934/dcds.2005.13.491 |
[4] |
Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 |
[5] |
Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 |
[6] |
Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021006 |
[7] |
Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011 |
[8] |
Ying Yang. Global classical solutions to two-dimensional chemotaxis-shallow water system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2625-2643. doi: 10.3934/dcdsb.2020198 |
[9] |
Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 |
[10] |
Cicely K. Macnamara, Mark A. J. Chaplain. Spatio-temporal models of synthetic genetic oscillators. Mathematical Biosciences & Engineering, 2017, 14 (1) : 249-262. doi: 10.3934/mbe.2017016 |
[11] |
Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053 |
[12] |
Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817 |
[13] |
Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042 |
[14] |
Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009 |
[15] |
Andrea Tosin, Mattia Zanella. Uncertainty damping in kinetic traffic models by driver-assist controls. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021018 |
[16] |
Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044 |
[17] |
Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265 |
[18] |
Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027 |
[19] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[20] |
Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637 |
2019 Impact Factor: 0.649
Tools
Metrics
Other articles
by authors
[Back to Top]