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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math., 170 (2002), 119-179. 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.

[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.

[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.

[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.

[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 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[39]

J. Klein, Espaces variationnels et mécanique, Ann. Inst. Fourier, 12 (1962), 1-124. 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). doi: 10.1088/1751-8113/42/38/385209.

[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. doi: 10.1017/CBO9780511661839.

[43]

K. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147. 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. doi: 10.1142/9789812813749.

[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.

[46]

J. E. MarsdenS. PekarskyS. Shkoller and M. West, Variational methods, multisymplectic geometry and continuum mechanics, J. Geom. Phys., 38 (2001), 253-284. 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. doi: 10.1088/1751-8113/48/5/055206.

[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). doi: 10.1088/1751-8113/43/32/325204.

[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 (2001), 209-222.

[50]

E. Martínez, Lagrangian Mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259.

[51]

E. Martínez, Reduction in optimal control theory, Reports in Mathematical Physics, 53 (2004), 79-90. 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 (2005), 7145-7160. doi: 10.1088/0305-4470/38/32/005.

[53]

E. Martínez, Lie algebroids in classical mechanics and optimal control, SIGMA 3 (2007), Paper 050, 17 pp. doi: 10.3842/SIGMA.2007.050.

[54]

E. Martínez, Variational Calculus on Lie algebroids, ESAIM-COCV, 14 (2008), 356-380. doi: 10.1051/cocv:2007056.

[55]

E. MartínezT. Mestdag and W. Sarlet, Lie algebroid structures and Lagrangian systems on affine bundles, J. Geom. Phys., 44 (2002), 70-95. doi: 10.1016/S0393-0440(02)00114-6.

[56]

F. MunteanuA. M. Rey and M. Salgado, The Günther's formalism in classical field theory: momentum map and reduction, J. Math. Phys., 45 (2004), 1730-1751. doi: 10.1063/1.1688433.

[57]

A. Nijenhuis, Vector form brackets in Lie algebroids, Arch. Math. (Brno), 32 (1996), 317-323.

[58]

L. K. Norris, Generalized Symplectic Geometry on the Frame Bundle of a Manifold, Proc. Symposia in Pure Math., 54 (1993), 435-465.

[59]

C. Paufler and H. Romer, Geometry of Hamiltonean $n$-vector fields in multisymplectic field theory, J. Geom. Phys., 44 (2002), 52-69. doi: 10.1016/S0393-0440(02)00031-1.

[60]

M. Popescu and P. Popescu, Geometric objects defined by almost Lie structures, In Lie Algebroids, Banach Center Publications, 54 (2001), 217-233. doi: 10.4064/bc54-0-12.

[61]

G. Sardanashvily, Generalized Hamiltonian Formalism for Field Theory. Constraint Systems, World Scientific, Singapore, 1995. doi: 10.1142/9789812831484.

[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.

[63]

W. SarletT. Mestdag and E. Martínez, Lie algebroid structures on a class of affine bundles, J. Math. Phys., 43 (2002), 5654-5674. 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. doi: 10.1017/CBO9780511526411.

[65]

J. Śniatycki, Multisymplectic reduction for proper actions, Canadian Journal of Maths., 56 (2004), 638-654. doi: 10.4153/CJM-2004-029-8.

[66]

T. Strobl, Gravity from Lie algebroid morphisms, Comm. Math. Phys., 246 (2004), 475-502. 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.

show all references

References:
[1]

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

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math., 170 (2002), 119-179. 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.

[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.

[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.

[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.

[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 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[39]

J. Klein, Espaces variationnels et mécanique, Ann. Inst. Fourier, 12 (1962), 1-124. 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). doi: 10.1088/1751-8113/42/38/385209.

[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. doi: 10.1017/CBO9780511661839.

[43]

K. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147. 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. doi: 10.1142/9789812813749.

[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.

[46]

J. E. MarsdenS. PekarskyS. Shkoller and M. West, Variational methods, multisymplectic geometry and continuum mechanics, J. Geom. Phys., 38 (2001), 253-284. 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. doi: 10.1088/1751-8113/48/5/055206.

[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). doi: 10.1088/1751-8113/43/32/325204.

[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 (2001), 209-222.

[50]

E. Martínez, Lagrangian Mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320. doi: 10.1023/A:1011965919259.

[51]

E. Martínez, Reduction in optimal control theory, Reports in Mathematical Physics, 53 (2004), 79-90. 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 (2005), 7145-7160. doi: 10.1088/0305-4470/38/32/005.

[53]

E. Martínez, Lie algebroids in classical mechanics and optimal control, SIGMA 3 (2007), Paper 050, 17 pp. doi: 10.3842/SIGMA.2007.050.

[54]

E. Martínez, Variational Calculus on Lie algebroids, ESAIM-COCV, 14 (2008), 356-380. doi: 10.1051/cocv:2007056.

[55]

E. MartínezT. Mestdag and W. Sarlet, Lie algebroid structures and Lagrangian systems on affine bundles, J. Geom. Phys., 44 (2002), 70-95. doi: 10.1016/S0393-0440(02)00114-6.

[56]

F. MunteanuA. M. Rey and M. Salgado, The Günther's formalism in classical field theory: momentum map and reduction, J. Math. Phys., 45 (2004), 1730-1751. doi: 10.1063/1.1688433.

[57]

A. Nijenhuis, Vector form brackets in Lie algebroids, Arch. Math. (Brno), 32 (1996), 317-323.

[58]

L. K. Norris, Generalized Symplectic Geometry on the Frame Bundle of a Manifold, Proc. Symposia in Pure Math., 54 (1993), 435-465.

[59]

C. Paufler and H. Romer, Geometry of Hamiltonean $n$-vector fields in multisymplectic field theory, J. Geom. Phys., 44 (2002), 52-69. doi: 10.1016/S0393-0440(02)00031-1.

[60]

M. Popescu and P. Popescu, Geometric objects defined by almost Lie structures, In Lie Algebroids, Banach Center Publications, 54 (2001), 217-233. doi: 10.4064/bc54-0-12.

[61]

G. Sardanashvily, Generalized Hamiltonian Formalism for Field Theory. Constraint Systems, World Scientific, Singapore, 1995. doi: 10.1142/9789812831484.

[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.

[63]

W. SarletT. Mestdag and E. Martínez, Lie algebroid structures on a class of affine bundles, J. Math. Phys., 43 (2002), 5654-5674. 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. doi: 10.1017/CBO9780511526411.

[65]

J. Śniatycki, Multisymplectic reduction for proper actions, Canadian Journal of Maths., 56 (2004), 638-654. doi: 10.4153/CJM-2004-029-8.

[66]

T. Strobl, Gravity from Lie algebroid morphisms, Comm. Math. Phys., 246 (2004), 475-502. 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.

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