December  2015, 7(4): 395-429. doi: 10.3934/jgm.2015.7.395

Symmetry reduction, integrability and reconstruction in $k$-symplectic field theory

1. 

Departamento de Xeometría e Topoloxía, Universidade de Santiago de Compostela, Spain, Spain

2. 

Department of Mathematics, Ghent University, Krijgslaan 281, B-9000 Gent, Belgium

Received  April 2015 Revised  August 2015 Published  October 2015

We investigate the reduction process of a $k$-symplectic field theory whose Lagrangian is invariant under a symmetry group. We give explicit coordinate expressions of the resulting reduced partial differential equations, the so-called Lagrange-Poincaré field equations. We discuss two issues about reconstructing a solution from a given solution of the reduced equations. The first one is an interpretation of the integrability conditions, in terms of the curvatures of some connections. The second includes the introduction of the concept of a $k$-connection to provide a reconstruction method. We show that an invariant Lagrangian, under suitable regularity conditions, defines a `mechanical' $k$-connection.
Citation: L. Búa, T. Mestdag, M. Salgado. Symmetry reduction, integrability and reconstruction in $k$-symplectic field theory. Journal of Geometric Mechanics, 2015, 7 (4) : 395-429. doi: 10.3934/jgm.2015.7.395
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, (Second Edition),, Benjamin-Cummings Publishing Company, (1978).   Google Scholar

[2]

L. Búa, I. Bucataru and M. Salgado, Symmetries, Newtonoid vector fields and conservation laws in the Lagrangian k-symplectic formalism,, Rev. Math. Phys., 24 (2012).  doi: 10.1142/S0129055X12500304.  Google Scholar

[3]

M. Castrillón López, P. L. García and C. Rodrigo, Euler-Poincaré reduction in principal bundles by a subgroup of the structure group,, J. Geom. Phys., 74 (2013), 352.  doi: 10.1016/j.geomphys.2013.08.008.  Google Scholar

[4]

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

[5]

J. F. Carinena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order field theories,, Differential Geometry and its Applications, 1 (1991), 345.  doi: 10.1016/0926-2245(91)90013-Y.  Google Scholar

[6]

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

[7]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometry of Lagrangian first-order classical field theories,, Forts. Phys., 44 (1996), 235.  doi: 10.1002/prop.2190440304.  Google Scholar

[8]

M. Crampin and T. Mestdag, Reduction and reconstruction aspects of second-order dynamical systems with symmetry,, Acta Appl. Math., 105 (2009), 241.  doi: 10.1007/s10440-008-9274-7.  Google Scholar

[9]

M. Crampin and F. A. E. Pirani, Applicable Differential Geometry,, London Mathematical Society Lecture Note Series, (1986).   Google Scholar

[10]

M. de Leon and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics,, North-Holland Mathematics Studies, (1989).   Google Scholar

[11]

D. C. P. Ellis, F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Lagrange-Poincaré field equations,, J. Geom. Phys., 61 (2011), 2120.  doi: 10.1016/j.geomphys.2011.06.007.  Google Scholar

[12]

D. C. P. Ellis, F. Gay-Balmaz, D. D. Holm, V. Putkaradze and T. S. Ratiu, Symmetry Reduced Dynamics of Charged Molecular Strands,, Arch. Ration. Mech. Anal., 197 (2010), 811.  doi: 10.1007/s00205-010-0305-y.  Google Scholar

[13]

R. Ghanam, G. Thompson and E. J. Miller, Variationality of four-dimensional Lie group Connections,, Journal of Lie Theory, 14 (2004), 395.   Google Scholar

[14]

M. J. Gotay, J. Isenberg, J. E. Marsden and R. Montgomery, Momentum Maps and Classical Relativistic Fields, Part I: Covariant Field Theory,, , (2004).   Google Scholar

[15]

C. Günther, The polysymplectic Hamiltonian formalism in field theory and calculus of variations I: The local case,, J. Differential Geom., 25 (1987), 23.   Google Scholar

[16]

F. Hélein and J. C. Wood, Harmonic maps,, In D. Krupka and D.J. Saunders, 1213 (2008), 417.  doi: 10.1016/B978-044452833-9.50009-7.  Google Scholar

[17]

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

[18]

J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories,, Lecture Notes in Physics, (1979).   Google Scholar

[19]

D. Krupka, Lagrange theory in fibered manifolds,, Rep. Math. Phys., 2 (1971), 121.  doi: 10.1016/0034-4877(71)90025-5.  Google Scholar

[20]

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

[21]

J. C. Marrero, N. Román-Roy, M. Salgado and S. Vilariño, Reduction of polysymplectic manifolds,, J. Phys. A: Math. Theor., 48 (2015).  doi: 10.1088/1751-8113/48/5/055206.  Google Scholar

[22]

T. Mestdag, A Lie algebroid approach to Lagrangian systems with symmetry,, In J. Bures et al (eds.), (2005), 523.   Google Scholar

[23]

T. Mestdag and M. Crampin, Invariant Lagrangians, mechanical connections and the Lagrange-Poincaré equations,, J. Phys. A: Math. Theor., 41 (2008).  doi: 10.1088/1751-8113/41/34/344015.  Google Scholar

[24]

M. McLean and L. K. Norris, Covariant field theory on frame bundles of fibered manifolds,, J. Math. Phys., 41 (2000), 6808.  doi: 10.1063/1.1288797.  Google Scholar

[25]

F. Munteanu, A. M. Rey and M. Salgado, The Günther's formalism in classical field theory: Momentum map and reduction,, J. Math. Phys., 45 (2004), 1730.  doi: 10.1063/1.1688433.  Google Scholar

[26]

P. J. Olver, Applications of Lie Groups to Differential Equations,, Springer-Verlag, (1986).  doi: 10.1007/978-1-4684-0274-2.  Google Scholar

[27]

N. Román-Roy, M. Salgado and S. Vilariño, Symmetries and Conservation Laws in Günter k-symplectic formalism of Field Theory,, Reviews in Mathematical Physics, 19 (2007), 1117.  doi: 10.1142/S0129055X07003188.  Google Scholar

[28]

N. Román-Roy, M. Salgado and S. Vilariño, SOPDEs and nonlinear connections,, Publ. Math. (Debrecen), 78 (2011), 297.  doi: 10.5486/PMD.2011.4631.  Google Scholar

[29]

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

[30]

D. J. Saunders, The Geometry of Jet Bundles,, Cambridge University Press, (1989).  doi: 10.1017/CBO9780511526411.  Google Scholar

[31]

J. Vankerschaver, Euler-Poincaré reduction for discrete field theories,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2712419.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, (Second Edition),, Benjamin-Cummings Publishing Company, (1978).   Google Scholar

[2]

L. Búa, I. Bucataru and M. Salgado, Symmetries, Newtonoid vector fields and conservation laws in the Lagrangian k-symplectic formalism,, Rev. Math. Phys., 24 (2012).  doi: 10.1142/S0129055X12500304.  Google Scholar

[3]

M. Castrillón López, P. L. García and C. Rodrigo, Euler-Poincaré reduction in principal bundles by a subgroup of the structure group,, J. Geom. Phys., 74 (2013), 352.  doi: 10.1016/j.geomphys.2013.08.008.  Google Scholar

[4]

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

[5]

J. F. Carinena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order field theories,, Differential Geometry and its Applications, 1 (1991), 345.  doi: 10.1016/0926-2245(91)90013-Y.  Google Scholar

[6]

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

[7]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometry of Lagrangian first-order classical field theories,, Forts. Phys., 44 (1996), 235.  doi: 10.1002/prop.2190440304.  Google Scholar

[8]

M. Crampin and T. Mestdag, Reduction and reconstruction aspects of second-order dynamical systems with symmetry,, Acta Appl. Math., 105 (2009), 241.  doi: 10.1007/s10440-008-9274-7.  Google Scholar

[9]

M. Crampin and F. A. E. Pirani, Applicable Differential Geometry,, London Mathematical Society Lecture Note Series, (1986).   Google Scholar

[10]

M. de Leon and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics,, North-Holland Mathematics Studies, (1989).   Google Scholar

[11]

D. C. P. Ellis, F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Lagrange-Poincaré field equations,, J. Geom. Phys., 61 (2011), 2120.  doi: 10.1016/j.geomphys.2011.06.007.  Google Scholar

[12]

D. C. P. Ellis, F. Gay-Balmaz, D. D. Holm, V. Putkaradze and T. S. Ratiu, Symmetry Reduced Dynamics of Charged Molecular Strands,, Arch. Ration. Mech. Anal., 197 (2010), 811.  doi: 10.1007/s00205-010-0305-y.  Google Scholar

[13]

R. Ghanam, G. Thompson and E. J. Miller, Variationality of four-dimensional Lie group Connections,, Journal of Lie Theory, 14 (2004), 395.   Google Scholar

[14]

M. J. Gotay, J. Isenberg, J. E. Marsden and R. Montgomery, Momentum Maps and Classical Relativistic Fields, Part I: Covariant Field Theory,, , (2004).   Google Scholar

[15]

C. Günther, The polysymplectic Hamiltonian formalism in field theory and calculus of variations I: The local case,, J. Differential Geom., 25 (1987), 23.   Google Scholar

[16]

F. Hélein and J. C. Wood, Harmonic maps,, In D. Krupka and D.J. Saunders, 1213 (2008), 417.  doi: 10.1016/B978-044452833-9.50009-7.  Google Scholar

[17]

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

[18]

J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories,, Lecture Notes in Physics, (1979).   Google Scholar

[19]

D. Krupka, Lagrange theory in fibered manifolds,, Rep. Math. Phys., 2 (1971), 121.  doi: 10.1016/0034-4877(71)90025-5.  Google Scholar

[20]

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

[21]

J. C. Marrero, N. Román-Roy, M. Salgado and S. Vilariño, Reduction of polysymplectic manifolds,, J. Phys. A: Math. Theor., 48 (2015).  doi: 10.1088/1751-8113/48/5/055206.  Google Scholar

[22]

T. Mestdag, A Lie algebroid approach to Lagrangian systems with symmetry,, In J. Bures et al (eds.), (2005), 523.   Google Scholar

[23]

T. Mestdag and M. Crampin, Invariant Lagrangians, mechanical connections and the Lagrange-Poincaré equations,, J. Phys. A: Math. Theor., 41 (2008).  doi: 10.1088/1751-8113/41/34/344015.  Google Scholar

[24]

M. McLean and L. K. Norris, Covariant field theory on frame bundles of fibered manifolds,, J. Math. Phys., 41 (2000), 6808.  doi: 10.1063/1.1288797.  Google Scholar

[25]

F. Munteanu, A. M. Rey and M. Salgado, The Günther's formalism in classical field theory: Momentum map and reduction,, J. Math. Phys., 45 (2004), 1730.  doi: 10.1063/1.1688433.  Google Scholar

[26]

P. J. Olver, Applications of Lie Groups to Differential Equations,, Springer-Verlag, (1986).  doi: 10.1007/978-1-4684-0274-2.  Google Scholar

[27]

N. Román-Roy, M. Salgado and S. Vilariño, Symmetries and Conservation Laws in Günter k-symplectic formalism of Field Theory,, Reviews in Mathematical Physics, 19 (2007), 1117.  doi: 10.1142/S0129055X07003188.  Google Scholar

[28]

N. Román-Roy, M. Salgado and S. Vilariño, SOPDEs and nonlinear connections,, Publ. Math. (Debrecen), 78 (2011), 297.  doi: 10.5486/PMD.2011.4631.  Google Scholar

[29]

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

[30]

D. J. Saunders, The Geometry of Jet Bundles,, Cambridge University Press, (1989).  doi: 10.1017/CBO9780511526411.  Google Scholar

[31]

J. Vankerschaver, Euler-Poincaré reduction for discrete field theories,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2712419.  Google Scholar

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