Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 100, 25 pages      math-ph/0506022      https://doi.org/10.3842/SIGMA.2009.100

Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories

Narciso Román-Roy
Dept. Matemática Aplicada IV, Edificio C-3, Campus Norte UPC, C/ Jordi Girona 1, E-08034 Barcelona, Spain

Received July 02, 2009, in final form October 30, 2009; Published online November 06, 2009

Abstract
This review paper is devoted to presenting the standard multisymplectic formulation for describing geometrically classical field theories, both the regular and singular cases. First, the main features of the Lagrangian formalism are revisited and, second, the Hamiltonian formalism is constructed using Hamiltonian sections. In both cases, the variational principles leading to the Euler-Lagrange and the Hamilton-De Donder-Weyl equations, respectively, are stated, and these field equations are given in different but equivalent geometrical ways in each formalism. Finally, both are unified in a new formulation (which has been developed in the last years), following the original ideas of Rusk and Skinner for mechanical systems.

Key words: classical field theories; Lagrangian and Hamiltonian formalisms; fiber bundles; multisymplectic manifolds.

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References

  1. Aldaya V., de Azcárraga J.A., Variational principles on rth order jets of fibre bundles in field theory, J. Math. Phys. 19 (1978), 1869-1875.
  2. Aldaya V., de Azcárraga J.A., Higher-order Hamiltonian formalism in field theory, J. Phys. A: Math. Gen. 13 (1980), 2545-2551.
  3. Aldaya V., de Azcárraga J.A., Geometric formulation of classical mechanics and field theory, Riv. Nuovo Cimento 3 (1980), no. 10, 1-66.
  4. Awane A., k-symplectic structures, J. Math. Phys. 32 (1992), 4046-4052.
  5. Awane A., G-espaces k-symplectiques homogènes, J. Geom. Phys. 13 (1994), 139-157.
  6. Awane A., Goze M., Pfaffian systems, k-symplectic systems, Kluwer Academic Publishers, Dordrecht, 2000.
  7. Barbero-Liñán M., Echeverría-Enríquez A., Martín de Diego D., Muñoz-Lecanda M.C., Román-Roy N., Unified formalism for nonautonomous mechanical systems, J. Math. Phys. 49 (2008), 062902, 14 pages, arXiv:0803.4085.
  8. Bashkirov D., BV quantization of covariant (polysymplectic) Hamiltonian field theory, Int. J. Geom. Methods Mod. Phys. 1 (2004), 233-252, hep-th/0403263.
  9. Betounes D.E., Extension of the classical Cartan-form, Phys. Rev. D 29 (1984), 599-606.
  10. Binz E., de León M., Martín de Diego D., Socolescu D., Nonholonomic constraints in classical field theories, Rep. Math. Phys. 49 (2002), 151-166, math-ph/0201038.
  11. Binz E., Sniatycki J., Fisher H., Geometry of classical fields, North-Holland Mathematics Studies, Vol. 154, North-Holland Publishing Co., Amsterdam, 1988.
  12. Blaga A.M., The prequantization of T1kRn, in Differential Geometry and Its Applications, Proc. Conf. in honour Leonard Euler (Olomouc, 2007), World Sci. Publ., Hackensack, NJ, 2008, 217-222.
  13. Campos C.M., de León M., Martín de Diego D., Vankerschaver J., Unambiguous formalism for higher-order Lagrangian field theories, J. Phys. A Math. Theor., to appear, arXiv:0906.0389.
  14. Cantrijn F., Ibort L.A., de León M., Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Politec. Torino 54 (1996), 225-236.
  15. Cantrijn F., Ibort L.A., de León M., On the geometry of multisymplectic manifolds, J. Austral. Math. Soc. Ser. A 66 (1999), 303-330.
  16. Cariñena J.F., Crampin M., Ibort L.A., On the multisymplectic formalism for first order field theories, Differential Geom. Appl. 1 (1991), 345-374.
  17. Castrillón-López M., García-Pérez P.L., Ratiu T.S., Euler-Poincaré reduction on principal bundles, Lett. Math. Phys. 58 (2001), 167-180.
  18. Castrillón-López M., Marsden J.E., Some remarks on Lagrangian and Poisson reduction for field theories, J. Geom. Phys. 48 (2003), 52-83.
  19. Castrillón-López M., Ratiu T.S., Shkoller S., Reduction in principal fiber bundles: covariant Euler-Poincaré equations, Proc. Amer. Math. Soc. 128 (2000), 2155-2164, math.DG/9908102.
  20. Cortés J., Martínez S., Cantrijn F., Skinner-Rusk approach to time-dependent mechanics, Phys. Lett. A 300 (2002), 250-258, math-ph/0203045.
  21. Dedecker P., On the generalization of symplectic geometry to multiple integrals in the calculus of variations, in Differential Geometrical Methods in Mathematical Physics (Proc. Sympos., Univ. Bonn, Bonn, 1975) Lecture Notes in Math., Vol. 570, Springer, Berlin, 1977, 395-456.
  22. Dedecker P., Problèmes variationnels dégénérés, C.R. Acad. Sci. Paris Sér. A-B 286 (1978), A547-A550.
  23. Echeverría-Enríquez A., de León M., Muñoz-Lecanda M.C., Román-Roy N., Extended Hamiltonian systems in multisymplectic field theories, J. Math. Phys. 48 (2007), 112901, 30 pages, math-ph/0506003.
  24. Echeverría-Enríquez A., López C., Marín-Solano J., Muñoz-Lecanda M.C., Román-Roy N., Lagrangian-Hamiltonian unified formalism for field theory, J. Math. Phys. 45 (2004), 360-380, math-ph/0212002.
  25. Echeverría-Enríquez A., Muñoz-Lecanda M.C., Variational calculus in several variables: a Hamiltonian approach, Ann. Inst. Henri Poincaré Phys. Theór. 56 (1992), 27-47.
  26. Echeverría-Enríquez A., Muñoz-Lecanda M.C., Román-Roy N., Geometry of Lagrangian first-order classical field theories, Fortschr. Phys. 44 (1996), 235-280, dg-ga/9505004.
  27. Echeverría-Enríquez A., Muñoz-Lecanda M.C., Román-Roy N., Multivector fields and connections: setting Lagrangian equations in field theories, J. Math. Phys. 39 (1998), 4578-4603, dg-ga/9707001.
  28. Echeverría-Enríquez A., Muñoz-Lecanda M.C., Román-Roy N., Multivector field formulation of Hamiltonian field theories: equations and symmetries, J. Phys. A: Math. Gen. 32 (1999), 8461-8484, math-ph/9907007.
  29. Echeverría-Enríquez A., Muñoz-Lecanda M.C., Román-Roy N., On the Multimomentum bundles and the Legendre Maps in field theories, Rep. Math. Phys. 45 (2000), 85-105, math-ph/9904007.
  30. Echeverría-Enríquez A., Muñoz-Lecanda M.C., Román-Roy N., Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys. 41 (2000), 7402-7444, math-ph/0004005.
  31. Ferraris M., Francaviglia M., Applications of the Poincaré-Cartan form in higher order field theories, in Differential Geometry and Its Applications (Brno, 1986), Math. Appl. (East European Ser.), Vol. 27, Reidel, Dordrecht, 1987, 31-52.
  32. Ferraris M., Francaviglia M., Intrinsic ADM formalism for generally covariant higher-order field theories, Atti Sem. Mat. Fis. Univ. Modena 37 (1989), 61-78.
  33. Forger M., Gomes L., Multisymplectic and polysymplectic structures on fiber bundles, arXiv:0708.1586.
  34. Forger M., Paufler C., Römer H., A general construction of Poisson brackets on exact multisymplectic manifolds, Rep. Math. Phys. 51 (2003), 187-195, math-ph/0208037.
  35. Forger M., Paufler C., Römer H., The Poisson bracket for Poisson forms in multisymplectic field theory, Rev. Math. Phys. 15 (2003), 705-743, math-ph/0202043.
  36. Forger M., Paufler C., Römer H., Hamiltonian multivector fields and Poisson forms in multisymplectic field theory, J. Math. Phys. 46 (2005), 112903, 29 pages, math-ph/0407057.
  37. García P.L., The Poincaré-Cartan invariant in the calculus of variations, Symposia Mathematica, Vol. 14 (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973), Academic Press, London, 1974, 219-246.
  38. García P.L., Muñoz J., On the geometrical structure of higher order variational calculus, Proceedings of the IUTAM-ISIMM symposium on modern developments in analytical mechanics, Vol. I (Torino, 1982), Editors M. Francaviglia and A. Lichnerowicz, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 117 (1983), suppl. 1, 127-147.
  39. Giachetta G., Mangiarotti L., Sardanashvily G., New Lagrangian and Hamiltonian methods in field theory, World Scientific Publishing Co., Inc., River Edge, NJ, 1997.
  40. Goldschmidt H., Sternberg S., The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier (Grenoble) 23 (1973), 203-267.
  41. Gotay M.J., A multisymplectic framework for classical field theory and the calculus of variations. I. Covariant Hamiltonian formalism, in Mechanics, Analysis and Geometry: 200 Years after Lagrange, Editor M. Francaviglia, North-Holland, Amsterdam, 1991, 203-235.
  42. Gotay M.J., A multisymplectic framework for classical field theory and the calculus of variations. II. Space + time decomposition, Differential Geom. Appl. 1 (1991), 375-390.
  43. Gotay M.J., Isenberg J., Marsden J.E., Momentum maps and classical relativistic fields. I. Covariant theory, MSRI Preprints, 1999.
  44. Gotay M.J., Isenberg J., Marsden J.E., Momentum maps and classical relativistic fields. II. Canonical analysis of field theories, MSRI Preprints, 1999.
  45. Gràcia X., Martín R., Geometric aspects of time-dependent singular differential equations, Int. J. Geom. Methods Mod. Phys. 2 (2005), 597-618.
  46. Günther C., The polysymplectic Hamiltonian formalism in the field theory and the calculus of variations. I. The local case, J. Differential Geom. 25 (1987), 23-53.
  47. Hélein F., Kouneiher J., Finite dimensional Hamiltonian formalism for gauge and quantum field theories, J. Math. Phys. 43 (2002), 2306-2347, math-ph/0004020.
  48. Hélein F., Kouneiher J., Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage-Dedecker versus De Donder-Weyl, Adv. Theor. Math. Phys. 8 (2004), 565-601, math-ph/0401046.
  49. Ibort L.A., Multisymplectic manifolds: general aspects and particular situations, Proceedings of the IX Fall Workshop on Geometry and Physics (Vilanova i la Geltrú, 2000), Editors X. Gràcia, J. Marín-Solano, M.C. Muñoz-Lecanda and N. Román-Roy, Publ. R. Soc. Mat. Esp., Vol. 3, R. Soc. Mat. Esp., Madrid, 2001, 79-88.
  50. Kanatchikov I.V., From the Poincaré-Cartan form to a Gerstenhaber algebra of the Poisson brackets in field theory, in Quantization, coherent states, and complex structures (Bialowieza, 1994), Plenum Press, New York, 1995, 173-183, hep-th/9511039.
  51. Kanatchikov I.V., On field-theoretic generalizations of a Poisson algebra, Rep. Math. Phys. 40 (1997), 225-234, hep-th/9710069.
  52. Kanatchikov I.V., Canonical structure of classical field theory in the polymomentum phase space, Rep. Math. Phys. 41 (1998), 49-90, hep-th/9709229.
  53. Kanatchikov I.V., Toward the Born-Weyl quantization of fields, Internat. J. Theoret. Phys. 37 (1998), 333-342, quant-ph/9712058.
  54. Kanatchikov I.V., De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory, Rep. Math. Phys. 43 (1999), 157-170, hep-th/9810165.
  55. Kanatchikov I.V., Precanonical perspective in quantum gravity, Nuclear Phys. B Proc. Suppl. 88 (2000), 326-330, gr-qc/0004066.
  56. Kanatchikov I.V., Geometric (pre)quantization in the polysymplectic approach to field theory, in Differential geometry and its applications (Opava, 2001), Math. Publ., Vol. 3, Silesian Univ. Opava, Opava, 2001, 309-321, hep-th/0112263.
  57. Kanatchikov I.V., Precanonical quantization and the Schrödinger wave functional, Phys. Lett. A 283 (2001), 25-36, hep-th/0012084.
  58. Kanatchikov I.V., Precanonical quantum gravity: quantization without the space-time decomposition, Internat. J. Theoret. Phys. 40 (2001), 1121-1149, gr-qc/0012074.
  59. Kanatchikov I.V., Precanonical quantization of Yang-Mills fields and the functional Schrödinger representation, Rep. Math. Phys. 53 (2004), 181-193, hep-th/0301001.
  60. Kijowski J., A finite-dimensional canonical formalism in the classical field theory, Comm. Math. Phys. 30 (1973), 99-128.
  61. Kijowski J., Tulczyjew W.M., A symplectic framework for field theories, Lecture Notes in Physics, Vol. 170, Springer-Verlag, Berlin - New York, 1979.
  62. Kouranbaeva S., Shkoller S., A variational approach to second-order multisymplectic field theory, J. Geom. Phys. 35 (2000), 333-366, math.DG/9909100.
  63. Krupka D., On the higher order Hamilton theory in fibered spaces, in Proceedings of the Conference on Differential Geometry and Its Applications, Part 2, Univ. J. E. Purkyne, Brno, 1984, 167-183.
  64. Krupka D., Regular Lagrangians and Lepagean forms, in Differential Geometry and Its Applications (Brno, 1986), Math. Appl. (East European Ser.), Vol. 27, Reidel, Dordrecht, 1987, 111-148.
  65. Krupkova O., Hamiltonian field theory, J. Geom. Phys. 43 (2002), 93-132.
  66. Krupkova O., Smetanova D., On regularization of variational problems in first-order field theory, in Proceedings of the 20th Winter School "Geometry and Physics" (Srni, 2000), Rend. Circ. Mat. Palermo (2) Suppl. 66 (2001), 133-140.
  67. Krupkova O., Smetanova D., Legendre transformation for regularizable Lagrangians in field theory, Lett. Math. Phys. 58 (2002), 189-204, math-ph/0111004.
  68. Lawson J.K., A frame bundle generalization of multisymplectic geometries, Rep. Math. Phys. 45 (2000), 183-205, dg-ga/9706008.
  69. de León M., Marín-Solano J., Marrero J.C., A geometrical approach to classical field theories: a constraint algorithm for singular theories, in New Developments in Differential Geometry (Debrecen, 1994), Editors L. Tamassi and J. Szenthe, Math. Appl., Vol. 350, Kluwer Acad. Publ., Dordrecht, 1996, 291-312.
  70. de León M., Marín-Solano J., Marrero J.C., Muñoz-Lecanda M.C., Román-Roy N., Pre-multisymplectic constraint algorithm for field theories, Int. J. Geom. Methods Mod. Phys. 2 (2005), 839-871, math-ph/0506005.
  71. de León M., Marrero J.C., Martín de Diego D., A new geometrical setting for classical field theories, in Classical and Quantum Integrability (Warsaw, 2001), Banach Center Pub., Vol. 59, Inst. of Math., Polish Acad. Sci., Warsawa, 2003, 189-209, math-ph/0202012.
  72. de León M., Martín de Diego D., Salgado M., Vilariño S., k-symplectic formalism on Lie algebroids, J. Phys. A: Math. Theor. 42 (2009), 385209, 31 pages, arXiv:0905.4585.
  73. de León M., Martín de Diego D., Santamaría-Merino A., Tulczyjew's triples and Lagrangian submanifolds in classical field theories, in Applied Differential Geometry and Mechanics, Editors W. Sarlet and F. Cantrijn, Univ. of Gent, Gent, Academia Press, 2003, 21-47, math-ph/0302026.
  74. de León M., Martín de Diego D., Santamaría-Merino A., Symmetries in classical field theories, Int. J. Geom. Methods Mod. Phys. 1 (2004), 651-710, math-ph/0404013.
  75. de León M., Merino E., Oubiña J.A., Rodrigues P., Salgado M., Hamiltonian systems on k-cosymplectic manifolds, J. Math. Phys. 39 (1998), 876-893.
  76. de León M., Merino E., Salgado M., k-cosymplectic manifolds and Lagrangian field theories, J. Math. Phys. 42 (2001), 2092-2104.
  77. Marsden J.E., Patrick G.W., Shkoller S., Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys. 199 (1998), 351-395, math.DG/9807080.
  78. Marsden J.E., Patrick G.W., Shkoller S., West M., Variational methods, multisymplectic geometry and continuum mechanics, J. Geom. Phys. 38 (2001), 253-284, math.DG/0005034.
  79. Marsden J.E., Shkoller S., Multisymplectic geometry, covariant Hamiltonians and water waves, Math. Proc. Cambridge Philos. Soc. 125 (1999), 553-575, math.DG/9807086.
  80. Marsden J.E., Weinstein A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121-130.
  81. Marsden J.E., Weinstein A., Some comments on the history, theory, and applications of symplectic reduction, in Quantization of Singular Symplectic Quotiens, Editors N. Landsman, M. Pflaum and M. Schlichenmanier, Progr. Math., Vol. 198, Birkhäuser, Basel, 2001, 1-19.
  82. Martínez E., Classical field theory on Lie algebroids: multisymplectic formalism, math.DG/0411352.
  83. Martínez E., Classical field theories on Lie algebroids: variational aspects, J. Phys. A: Math. Gen. 38 (2005), 7145-7160, math-dg/0410551.
  84. Munteanu F., Rey A.M., Salgado M., The Günther's formalism in classical field theory: momentum map and reduction, J. Math. Phys. 45 (2004), 1730-1751.
  85. McLean M., Norris L.K., Covariant field theory on frame bundles of fibered manifolds, J. Math. Phys. 41 (2000), 6808-6823.
  86. Norris L.K., Generalized symplectic geometry on the frame bundle of a manifold, in Differential Geometry: Geometry in Mathematical Physics and Related Topics (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., Vol. 54, Part 2, Amer. Math. Soc., Providence, RI, 1993, 435-465.
  87. Norris L.K., n-symplectic algebra of observables in covariant Lagrangian field theory, J. Math. Phys. 42 (2001), 4827-4845.
  88. Paufler C., Römer H., Geometry of Hamiltonian n-vector fields in multisymplectic field theory, J. Geom. Phys. 44 (2002), 52-69, math-ph/0102008.
  89. Puta M., Chirici S., Merino E., On the prequantization of (T1k)*Rn, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 44(92) (2001), 277-284.
  90. Sardanashvily G., Multimomentum canonical quantization of fields, Hadronic J. 17 (1994), 227-245.
  91. Sardanashvily G., Generalized Hamiltonian formalism for field theory. Constraint systems, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
  92. Saunders D.J., The Geometry of jet bundles, London Mathematical Society Lecture Note Series, Vol. 142, Cambridge University Press, Cambridge, 1989.
  93. Saunders D.J., Crampin M., On the Legendre map in higher-order field theories, J. Phys. A: Math. Gen. 23 (1990), 3169-3182.
  94. Skinner R., Rusk R., Generalized Hamiltonian dynamics. I. Formulation on T*QTQ, J. Math. Phys. 24 (1983), 2589-2594.
  95. Vankerschaver J., Cantrijn F., de León M., Martín de Diego D., Geometric aspects of nonholonomic field theories, Rep. Math. Phys. 56 (2005), 387-411, math-ph/0506010.
  96. Vankerschaver J., The momentum map for nonholonomic field theories with symmetry, Int. J. Geom. Methods Mod. Phys. 2 (2005), 1029-1041, math-ph/0507059.
  97. Vankerschaver J., Euler-Poincaré reduction for discrete field theories, J. Math. Phys. 48 (2007), 032902, 17 pages, math-ph/0606033.
  98. Vankerschaver J., Martín de Diego D., Symmetry aspects of nonholonomic field theories, J. Phys. A: Math. Theor. 41 (2008), 035401, 17 pages, arXiv:0712.2272.

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