March  2018, 10(1): 93-138. doi: 10.3934/jgm.2018004

Classical field theory on Lie algebroids: Multisymplectic formalism

IUMA and Department of Applied Mathematics, University of Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain

Received  August 2017 Revised  November 2017 Published  December 2017

Fund Project: Partial financial support from MINECO (Spain) grant MTM2015-64166-C2-1-P, and from Gobierno de Aragón (Spain) grant DGA-E24/1 is acknowledged

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.

Citation: Eduardo Martínez. Classical field theory on Lie algebroids: Multisymplectic formalism. Journal of Geometric Mechanics, 2018, 10 (1) : 93-138. doi: 10.3934/jgm.2018004
References:
[1]

E. Binz, J. Śniatycki and H. Fisher, The Geometry of Classical fields, North Holland, Amsterdam, 1988. Google Scholar

[2]

M. BojowaldA. Kotov and T. Strobl, Lie algebroid morphisms, Poisson Sigma Models, and off-shell closed gauge symmetries, J. Geom. Phys., 54 (2005), 400-426. doi: 10.1016/j.geomphys.2004.11.002. Google Scholar

[3]

T. J. Bridges and S. Reich, Multi-symplectic integrators: Numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001), 184-193. doi: 10.1016/S0375-9601(01)00294-8. Google Scholar

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A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Amer. Math. Soc., Providence, RI, 1999; xiv+184 pp. Google Scholar

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F. Cantrijn and B. Langerock, Generalised connections over a vector bundle map, Differential Geom. Appl., 18 (2003), 295-317. doi: 10.1016/S0926-2245(02)00164-X. Google Scholar

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J. F. CariñenaM. 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. Google Scholar

[7]

M. Castrillón López and T. S. Ratiu, Reduction in principal bundles: Covariant Lagrange-Poincaré equations, Comm. Math. Phys., 236 (2003), 223-250. doi: 10.1007/s00220-003-0797-5. Google Scholar

[8]

M. Castrillón LópezP. L. García-Pérez and T. S. Ratiu, Euler-Poincaré reduction on principal bundles, Lett. Math. Phys., 58 (2001), 167-180. doi: 10.1023/A:1013303320765. Google Scholar

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M. Castrillón LópezT. S. Ratiu and S. Shkoller, Reduction in principal fiber bundles: Covariant Euler-Poincaré equations, Proc. Amer. Math. Soc., 128 (2000), 2155-2164. doi: 10.1090/S0002-9939-99-05304-6. Google Scholar

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J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids, IMA Journal of Mathematical Control and Information, 21 (2004), 457-492. doi: 10.1093/imamci/21.4.457. Google Scholar

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T. De Donder, Théorie invariantive du calcul des variations, Bull. Acad. de Belg., 1929.Google Scholar

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M. de LeónJ. Marín-Solano and 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 (1996), 291-312. Google Scholar

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M. de LeónJ. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38 (2005), R241-R308. doi: 10.1088/0305-4470/38/24/R01. Google Scholar

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A. Echeverría-EnríquezM. 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. Google Scholar

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J. Grabowski and P. Urbanski, Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids, Ann. Global Anal. Geom., 15 (1997), 447-486. doi: 10.1023/A:1006519730920. Google Scholar

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show all references

References:
[1]

E. Binz, J. Śniatycki and H. Fisher, The Geometry of Classical fields, North Holland, Amsterdam, 1988. Google Scholar

[2]

M. BojowaldA. Kotov and T. Strobl, Lie algebroid morphisms, Poisson Sigma Models, and off-shell closed gauge symmetries, J. Geom. Phys., 54 (2005), 400-426. doi: 10.1016/j.geomphys.2004.11.002. Google Scholar

[3]

T. J. Bridges and S. Reich, Multi-symplectic integrators: Numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001), 184-193. doi: 10.1016/S0375-9601(01)00294-8. Google Scholar

[4]

A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Amer. Math. Soc., Providence, RI, 1999; xiv+184 pp. Google Scholar

[5]

F. Cantrijn and B. Langerock, Generalised connections over a vector bundle map, Differential Geom. Appl., 18 (2003), 295-317. doi: 10.1016/S0926-2245(02)00164-X. Google Scholar

[6]

J. F. CariñenaM. 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. Google Scholar

[7]

M. Castrillón López and T. S. Ratiu, Reduction in principal bundles: Covariant Lagrange-Poincaré equations, Comm. Math. Phys., 236 (2003), 223-250. doi: 10.1007/s00220-003-0797-5. Google Scholar

[8]

M. Castrillón LópezP. L. García-Pérez and T. S. Ratiu, Euler-Poincaré reduction on principal bundles, Lett. Math. Phys., 58 (2001), 167-180. doi: 10.1023/A:1013303320765. Google Scholar

[9]

M. Castrillón LópezT. S. Ratiu and S. Shkoller, Reduction in principal fiber bundles: Covariant Euler-Poincaré equations, Proc. Amer. Math. Soc., 128 (2000), 2155-2164. doi: 10.1090/S0002-9939-99-05304-6. Google Scholar

[10]

H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc. 152 (2001), x+108 pp. doi: 10.1090/memo/0722. Google Scholar

[11]

J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids, IMA Journal of Mathematical Control and Information, 21 (2004), 457-492. doi: 10.1093/imamci/21.4.457. Google Scholar

[12]

T. De Donder, Théorie invariantive du calcul des variations, Bull. Acad. de Belg., 1929.Google Scholar

[13]

M. de LeónJ. Marín-Solano and 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 (1996), 291-312. Google Scholar

[14]

M. de LeónJ. C. Marrero and 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 (2003), 189-209. doi: 10.4064/bc59-0-10. Google Scholar

[15]

M. de LeónJ. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen., 38 (2005), R241-R308. doi: 10.1088/0305-4470/38/24/R01. Google Scholar

[16]

M. de LeónE. Merino and M. Salgado, k-cosymplectic manifolds and Lagrangian field theories, J. Math. Phys., 42 (2001), 2092-2104. doi: 10.1063/1.1360997. Google Scholar

[17]

M. de LeónE. MerinoJ. A. OubiñaP. R. Rodrigues and M. Salgado, Hamiltonian Systems on $k$-cosymplectic Manifolds, J. Math. Phys., 39 (1998), 876-893. doi: 10.1063/1.532358. Google Scholar

[18]

A. Echeverría-EnríquezM. 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. Google Scholar

[19]

A. Echeverría-EnríquezM. 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. Google Scholar

[20]

A. Echeverría-EnríquezM. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of hamiltonian field theories: Equations and symmetries, J. Math. Phys., 41 (2000), 7402-7444. doi: 10.1088/0305-4470/32/48/309. Google Scholar

[21]

A. Echeverría-EnríquezM. C. Muñoz-Lecanda and N. Román-Roy, Geometry of multisymplectic hamiltonian first-order field theories, J. Math. Phys., 41 (2000), 7402-7444. doi: 10.1063/1.1308075. Google Scholar

[22]

A. Echeverría-EnríquezC. LópezJ. Marín-SolanoM. C. Muñoz-Lecanda and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for field theory, J. Math. Phys., 45 (2004), 360-380. doi: 10.1063/1.1628384. Google Scholar

[23]

R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math., 170 (2002), 119-179. doi: 10.1006/aima.2001.2070. Google Scholar

[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. Google Scholar

[25]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, New Lagrangian and Hamiltonian Methods in Field Theory, World Scientific Pub. Co., Singapore, 1997. doi: 10.1142/2199. Google Scholar

[26]

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. Google Scholar

[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, (1991), 203-235. Google Scholar

[28]

M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields. Part I: Covariant field theory, arXiv: physics/9801019Google Scholar

[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/0411032Google Scholar

[30]

J. Grabowski and P. Urbanski, Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids, Ann. Global Anal. Geom., 15 (1997), 447-486. doi: 10.1023/A:1006519730920. Google Scholar

[31]

K. GrabowskaJ. Grabowski and P. Rubanski, Lie brackets on affine bundles, Annals of Global Analysis and Geometry, 24 (2003), 101-130. doi: 10.1023/A:1024457728027. Google Scholar

[32]

C. Günther, The polysymplectic Hamiltonian formalism in field theory and calculus of variations I: The local case, J. Diff. Geom., 25 (1987), 23-53. doi: 10.4310/jdg/1214440723. Google Scholar

[33]

F. Hélein and J. Kouneiher, Finite dimensional Hamiltonian formalism for gauge and quantum field theories, J. Math. Phys., 43 (2002), 2306-2347. doi: 10.1063/1.1467710. Google Scholar

[34]

F. Hélein and J. Kouneiher, Covariant Hamiltonian formalism for the calculus of variations with several variables, Adv. Theor. Math. Phys., 8 (2004), 565-601. doi: 10.4310/ATMP.2004.v8.n3.a5. Google Scholar

[35]

P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. of Algebra, 129 (1990), 194-230. doi: 10.1016/0021-8693(90)90246-K. Google Scholar

[36]

D. IglesiasJ. C. MarreroE. Padrón and D. Sosa, Lagrangian submanifolds and dynamics on Lie affgebroids, Reports on Mathematical Physics, 57 (2006), 385-436. doi: 10.1016/S0034-4877(06)80029-7. Google Scholar

[37]

I. V. Kanatchikov, Canonical structure of Classical Field Theory in the polymomentum phase space, Rep. Math. Phys., 41 (1998), 49-90. doi: 10.1016/S0034-4877(98)80182-1. Google Scholar

[38]

J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories, Lect. Notes Phys., 170 Springer-Verlag, Berlin, 1979. Google Scholar

[39]

J. Klein, Espaces variationnels et mécanique, Ann. Inst. Fourier, 12 (1962), 1-124. doi: 10.5802/aif.120. Google Scholar

[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). doi: 10.1088/1751-8113/42/38/385209. Google Scholar

[41]

T. Lepage, Acad. Roy. Belgique. Bull. Cl. Sci., 22 (1936), 716-735.Google Scholar

[42]

K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Math. Soc. Lect. Note Series, 124 (Cambridge Univ. Press), 1987. doi: 10.1017/CBO9780511661839. Google Scholar

[43]

K. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147. doi: 10.1112/blms/27.2.97. Google Scholar

[44]

L. Mangiarotti and G. Sardanashvily, Connections in Classical and Quantum Field Theory, World Scientific Publishing (River Edge, NJ), 2000. doi: 10.1142/9789812813749. Google Scholar

[45]

J. E. MarsdenG. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395. doi: 10.1007/s002200050505. Google Scholar

[46]

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