Classical field theory on Lie algebroids: multisymplectic formalism. (English) Zbl 1406.58003
Summary: The jet formalism for classical field theories is extended to the setting of Lie algebroids. We define the analog of the concept of jet of a section of a bundle and we study some of the geometric structures of the jet manifold. When a Lagrangian function is given, we find the equations of motion in terms of a Cartan form canonically associated to the Lagrangian. The Hamiltonian formalism is also extended to this setting and we find the relation between the solutions of both formalism. When the first Lie algebroid is a tangent bundle, we give a variational description of the equations of motion. In addition to the standard case, our formalism includes as particular examples the case of systems with symmetry (covariant Euler-Poincaré and Lagrange Poincaré cases), variational problems for holomorphic maps, Sigma models or Chern-Simons theories. One of the advantages of our theory is that it is based in the existence of a multisymplectic form on a Lie algebroid.
MSC:
| 58A20 | Jets in global analysis |
| 70S05 | Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems |
| 49S05 | Variational principles of physics |
| 35F20 | Nonlinear first-order PDEs |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
| 53D42 | Symplectic field theory; contact homology |
Keywords:
Lie algebroids; jet bundles; Lagrangian field theory; Hamiltonian field theory; multisymplectic formsReferences:
| [1] | E. Binz, J. Śniatycki and H. Fisher, The Geometry of Classical fields, North Holland, Amsterdam, 1988. · Zbl 0675.53065 |
| [2] | M. Bojowald; A. Kotov; T. Strobl, Lie algebroid morphisms, Poisson Sigma Models, and off-shell closed gauge symmetries, J. Geom. Phys., 54, 400-426 (2005) · Zbl 1076.53102 · doi:10.1016/j.geomphys.2004.11.002 |
| [3] | T. J. Bridges; S. Reich, Multi-symplectic integrators: Numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284, 184-193 (2001) · Zbl 0984.37104 · doi:10.1016/S0375-9601(01)00294-8 |
| [4] | A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Amer. Math. Soc., Providence, RI, 1999; xiv+184 pp. · Zbl 1135.58300 |
| [5] | F. Cantrijn; B. Langerock, Generalised connections over a vector bundle map, Differential Geom. Appl., 18, 295-317 (2003) · Zbl 1036.53014 · doi:10.1016/S0926-2245(02)00164-X |
| [6] | J. F. Cariñena; M. Crampin; L. A. Ibort, On the multisymplectic formalism for first order field theories, Diff. Geom. Appl., 1, 345-374 (1991) · Zbl 0782.58057 · doi:10.1016/0926-2245(91)90013-Y |
| [7] | M. Castrillón López; T. S. Ratiu, Reduction in principal bundles: Covariant Lagrange-Poincaré equations, Comm. Math. Phys., 236, 223-250 (2003) · Zbl 1037.53056 · doi:10.1007/s00220-003-0797-5 |
| [8] | M. Castrillón López; P. L. García-Pérez; T. S. Ratiu, Euler-Poincaré reduction on principal bundles, Lett. Math. Phys., 58, 167-180 (2001) · Zbl 1020.58018 · doi:10.1023/A:1013303320765 |
| [9] | M. Castrillón López; T. S. Ratiu; S. Shkoller, Reduction in principal fiber bundles: Covariant Euler-Poincaré equations, Proc. Amer. Math. Soc., 128, 2155-2164 (2000) · Zbl 0967.53019 · doi:10.1090/S0002-9939-99-05304-6 |
| [10] | H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc. 152 (2001), x+108 pp. · Zbl 1193.37072 |
| [11] | J. Cortés; E. Martínez, Mechanical control systems on Lie algebroids, IMA Journal of Mathematical Control and Information, 21, 457-492 (2004) · Zbl 1106.93020 · doi:10.1093/imamci/21.4.457 |
| [12] | T. De Donder, Théorie invariantive du calcul des variations, Bull. Acad. de Belg., 1929. · JFM 55.0903.01 |
| [13] | M. de León; J. Marín-Solano; J. C. Marrero, A Geometrical approach to Classical Field Theories: A constraint algorithm for singular theories, In Proc. on New Developments in Differential geometry, L. Tamassi-J. Szenthe eds., Kluwer Acad. Press, 350, 291-312 (1996) · Zbl 0843.70010 |
| [14] | M. de León; J. C. Marrero; D. Martín de Diego, A new geometric setting for classical field theories, In Classical and Quantum Integrability, Banach Center Pub., Inst. of Math., Polish Acad. Sci., Warsawa, 59, 189-209 (2003) · Zbl 1036.70016 · doi:10.4064/bc59-0-10 |
| [15] | M. de León; J. C. Marrero; E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38, R241-R308 (2005) · Zbl 1181.37003 · doi:10.1088/0305-4470/38/24/R01 |
| [16] | M. de León; E. Merino; M. Salgado, k-cosymplectic manifolds and Lagrangian field theories, J. Math. Phys., 42, 2092-2104 (2001) · Zbl 1045.70016 · doi:10.1063/1.1360997 |
| [17] | M. de León; E. Merino; J. A. Oubiña; P. R. Rodrigues; M. Salgado, Hamiltonian Systems on \(k\)-cosymplectic Manifolds, J. Math. Phys., 39, 876-893 (1998) · Zbl 0910.53022 · doi:10.1063/1.532358 |
| [18] | A. Echeverría-Enríquez; M. C. Muñoz-Lecanda; N. Román-Roy, Geometry of Lagrangian first-order classical field theories, Forts. Phys., 44, 235-280 (1996) · Zbl 0964.58015 · doi:10.1002/prop.2190440304 |
| [19] | A. Echeverría-Enríquez; M. C. Muñoz-Lecanda; N. Román-Roy, Multivector Fields and Connections: Setting Lagrangian Equations in Field Theories, J. Math. Phys., 39, 4578-4603 (1998) · Zbl 0927.37054 · doi:10.1063/1.532525 |
| [20] | A. Echeverría-Enríquez; M. C. Muñoz-Lecanda; N. Román-Roy, Multivector field formulation of hamiltonian field theories: Equations and symmetries, J. Math. Phys., 41, 7402-7444 (2000) · Zbl 0982.70019 · doi:10.1088/0305-4470/32/48/309 |
| [21] | A. Echeverría-Enríquez; M. C. Muñoz-Lecanda; N. Román-Roy, Geometry of multisymplectic hamiltonian first-order field theories, J. Math. Phys., 41, 7402-7444 (2000) · Zbl 1007.70023 · doi:10.1063/1.1308075 |
| [22] | A. Echeverría-Enríquez; C. López; J. Marín-Solano; M. C. Muñoz-Lecanda; N. Román-Roy, Lagrangian-Hamiltonian unified formalism for field theory, J. Math. Phys., 45, 360-380 (2004) · Zbl 1070.70015 · doi:10.1063/1.1628384 |
| [23] | R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math., 170, 119-179 (2002) · Zbl 1007.22007 · doi:10.1006/aima.2001.2070 |
| [24] | P. L. García-Pérez, The Poincaré-Cartan invariant in the calculus of variations, Symp. Math., 14 (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973), Acad. Press, London, (1974), 219-246. · Zbl 0303.53040 |
| [25] | G. Giachetta, L. Mangiarotti and G. Sardanashvily, New Lagrangian and Hamiltonian Methods in Field Theory, World Scientific Pub. Co., Singapore, 1997. · Zbl 0913.58001 |
| [26] | H. Goldschmidt; S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier, 23, 203-267 (1973) · Zbl 0243.49011 · doi:10.5802/aif.451 |
| [27] | M. J. Gotay, 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, M. Francaviglia Ed., Elsevier Science Pub, 203-235 (1991) · Zbl 0741.70012 |
| [28] | M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields. Part I: Covariant field theory, arXiv: physics/9801019 |
| [29] | M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields. Part II: Canonical analysis of field theories, arXiv: math-ph/0411032 |
| [30] | J. Grabowski; P. Urbanski, Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids, Ann. Global Anal. Geom., 15, 447-486 (1997) · Zbl 0973.58006 · doi:10.1023/A:1006519730920 |
| [31] | K. Grabowska; J. Grabowski; P. Rubanski, Lie brackets on affine bundles, Annals of Global Analysis and Geometry, 24, 101-130 (2003) · Zbl 1044.53056 · doi:10.1023/A:1024457728027 |
| [32] | C. Günther, The polysymplectic Hamiltonian formalism in field theory and calculus of variations I: The local case, J. Diff. Geom., 25, 23-53 (1987) · Zbl 0771.53012 · doi:10.4310/jdg/1214440723 |
| [33] | F. Hélein; J. Kouneiher, Finite dimensional Hamiltonian formalism for gauge and quantum field theories, J. Math. Phys., 43, 2306-2347 (2002) · Zbl 1059.70019 · doi:10.1063/1.1467710 |
| [34] | F. Hélein; J. Kouneiher, Covariant Hamiltonian formalism for the calculus of variations with several variables, Adv. Theor. Math. Phys., 8, 565-601 (2004) · Zbl 1115.70017 · doi:10.4310/ATMP.2004.v8.n3.a5 |
| [35] | P. J. Higgins; K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. of Algebra, 129, 194-230 (1990) · Zbl 0696.22007 · doi:10.1016/0021-8693(90)90246-K |
| [36] | D. Iglesias; J. C. Marrero; E. Padrón; D. Sosa, Lagrangian submanifolds and dynamics on Lie affgebroids, Reports on Mathematical Physics, 57, 385-436 (2006) · Zbl 1143.53073 · doi:10.1016/S0034-4877(06)80029-7 |
| [37] | I. V. Kanatchikov, Canonical structure of Classical Field Theory in the polymomentum phase space, Rep. Math. Phys., 41, 49-90 (1998) · Zbl 0947.70020 · doi:10.1016/S0034-4877(98)80182-1 |
| [38] | J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories, Lect. Notes Phys., 170 Springer-Verlag, Berlin, 1979. · Zbl 0439.58002 |
| [39] | J. Klein, Espaces variationnels et mécanique, Ann. Inst. Fourier, 12, 1-124 (1962) · Zbl 0281.49026 · doi:10.5802/aif.120 |
| [40] | M. de León, D. Martín de Diego, M. Salgado and S. Vilariño. K-symplectic formalism on Lie algebroids J. Phys. A: Math and Theor. , 42 (2009), 385209 (31 pp). · Zbl 1189.70140 |
| [41] | T. Lepage, Acad. Roy. Belgique. Bull. Cl. Sci., 22 (1936), 716-735. |
| [42] | K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Math. Soc. Lect. Note Series, 124 (Cambridge Univ. Press), 1987. · Zbl 0683.53029 |
| [43] | K. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27, 97-147 (1995) · Zbl 0829.22001 · doi:10.1112/blms/27.2.97 |
| [44] | L. Mangiarotti and G. Sardanashvily, Connections in Classical and Quantum Field Theory, World Scientific Publishing (River Edge, NJ), 2000. · Zbl 1053.53022 |
| [45] | J. E. Marsden; G. W. Patrick; S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys., 199, 351-395 (1998) · Zbl 0951.70002 · doi:10.1007/s002200050505 |
| [46] | J. E. Marsden; S. Pekarsky; S. Shkoller; M. West, Variational methods, multisymplectic geometry and continuum mechanics, J. Geom. Phys., 38, 253-284 (2001) · Zbl 1007.74018 · doi:10.1016/S0393-0440(00)00066-8 |
| [47] | J. C. Marrero, N. Román-Roy, N. Salgado and S. Vilariño, Reduction of polysymplectic manifolds, J. Phys. A: Math. Theor. , 48 (2015), 055206, 43pp. · Zbl 1321.53094 |
| [48] | D. Martín de Diego and S. Vilariño, Reduced Classical Field Theories: K-cosymplectic formalism on Lie algebroids, J. Phys. A: Math and Theor. , 43 (2010), 325204 (32 pp). · Zbl 1248.58011 |
| [49] | E. Martínez, Geometric formulation of mechanics on Lie algebroids, In Procs. of the VIII Fall Workshop on Geometry and Physics, Medina del Campo 1999, Publicaciones de la RSME, 2, 209-222 (2001) · Zbl 1044.70007 |
| [50] | E. Martínez, Lagrangian Mechanics on Lie algebroids, Acta Appl. Math., 67, 295-320 (2001) · Zbl 1002.70013 · doi:10.1023/A:1011965919259 |
| [51] | E. Martínez, Reduction in optimal control theory, Reports in Mathematical Physics, 53, 79-90 (2004) · Zbl 1060.58012 · doi:10.1016/S0034-4877(04)90005-5 |
| [52] | E. Martínez, Classical field theory on Lie algebroids: Variational aspects, J. Phys. A: Math. Gen., 38, 7145-7160 (2005) · Zbl 1073.70024 · doi:10.1088/0305-4470/38/32/005 |
| [53] | E. Martínez, Lie algebroids in classical mechanics and optimal control, SIGMA3 (2007), Paper 050, 17 pp. · Zbl 1142.49023 |
| [54] | E. Martínez, Variational Calculus on Lie algebroids, ESAIM-COCV, 14, 356-380 (2008) · Zbl 1135.49027 · doi:10.1051/cocv:2007056 |
| [55] | E. Martínez; T. Mestdag; W. Sarlet, Lie algebroid structures and Lagrangian systems on affine bundles, J. Geom. Phys., 44, 70-95 (2002) · Zbl 1010.37032 · doi:10.1016/S0393-0440(02)00114-6 |
| [56] | F. Munteanu; A. M. Rey; M. Salgado, The Günther’s formalism in classical field theory: momentum map and reduction, J. Math. Phys., 45, 1730-1751 (2004) · Zbl 1071.70017 · doi:10.1063/1.1688433 |
| [57] | A. Nijenhuis, Vector form brackets in Lie algebroids, Arch. Math. (Brno), 32, 317-323 (1996) · Zbl 0881.17020 |
| [58] | L. K. Norris, Generalized Symplectic Geometry on the Frame Bundle of a Manifold, Proc. Symposia in Pure Math., 54, 435-465 (1993) · Zbl 0790.53033 |
| [59] | C. Paufler; H. Romer, Geometry of Hamiltonean \(n\)-vector fields in multisymplectic field theory, J. Geom. Phys., 44, 52-69 (2002) · Zbl 1051.70020 · doi:10.1016/S0393-0440(02)00031-1 |
| [60] | M. Popescu; P. Popescu, Geometric objects defined by almost Lie structures, In Lie Algebroids, Banach Center Publications, 54, 217-233 (2001) · Zbl 1001.53011 · doi:10.4064/bc54-0-12 |
| [61] | G. Sardanashvily, Generalized Hamiltonian Formalism for Field Theory. Constraint Systems, World Scientific, Singapore, 1995. · Zbl 0941.70500 |
| [62] | W. Sarlet, T. Mestdag and E. Martínez, Lagrangian equations on affine Lie algebroids, In: Differential geometry and its applications (Opava, 2001), 461-472, Math. Publ., 3, Silesian Univ. Opava, Opava, 2001. · Zbl 1109.70304 |
| [63] | W. Sarlet; T. Mestdag; E. Martínez, Lie algebroid structures on a class of affine bundles, J. Math. Phys., 43, 5654-5674 (2002) · Zbl 1060.58014 · doi:10.1063/1.1510958 |
| [64] | D. J. Saunders, The Geometry of Jet Bundles, London Math. Soc. Lect. Notes Ser. 142, Cambridge, Univ. Press, 1989. · Zbl 0665.58002 |
| [65] | J. Śniatycki, Multisymplectic reduction for proper actions, Canadian Journal of Maths., 56, 638-654 (2004) · Zbl 1073.70025 · doi:10.4153/CJM-2004-029-8 |
| [66] | T. Strobl, Gravity from Lie algebroid morphisms, Comm. Math. Phys., 246, 475-502 (2004) · Zbl 1062.53068 · doi:10.1007/s00220-003-1026-y |
| [67] | A. Weinstein, Lagrangian Mechanics and groupoids, In: Mechanics day (Waterloo, ON, 1992), Fields Institute Communications, 7, American Mathematical Society (1996), 207-231. · Zbl 0844.22007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
