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March  2020, 12(1): 1-23. doi: 10.3934/jgm.2020002

Constraint algorithm for singular field theories in the k-cosymplectic framework

Xavier Gràcia , , Xavier Rivas and  Narciso Román-Roy

Department of Mathematics, Universitat Politècnica de Catalunya, Campus Nord UPC, edifici C3, C. Jordi Girona, 1, 08034 Barcelona, Catalonia, Spain

* Corresponding author: Xavier Gràcia

Received  January 2019 Revised  July 2019 Published  January 2020

The aim of this paper is to develop a constraint algorithm for singular classical field theories in the framework of $ k $-cosymplectic geometry. Since these field theories are singular, we need to introduce the notion of $ k $-precosymplectic structure, which is a generalization of the $ k $-cosymplectic structure. Next $ k $-precosymplectic Hamiltonian systems are introduced in order to describe singular field theories, both in Lagrangian and Hamiltonian formalisms. Finally, we develop a constraint algorithm in order to find a submanifold where the existence of solutions of the field equations is ensured. The case of affine Lagrangians is studied as a relevant example.

Keywords: k-cosymplectic manifold, k-precosymplectic manifold, constraint algorithm, singular field theories, Hamiltonian formalism, Lagrangian formalism, affine Lagrangian.
Mathematics Subject Classification: Primary: 70H45; Secondary: 35R01, 53C15, 53D99, 70G45, 70S05.
Citation: Xavier Gràcia, Xavier Rivas, Narciso Román-Roy. Constraint algorithm for singular field theories in the k-cosymplectic framework. Journal of Geometric Mechanics, 2020, 12 (1) : 1-23. doi: 10.3934/jgm.2020002
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Addison-Wesley, California, 2nd edition, 1978. doi: 10.1090/chel/364.  Google Scholar

[2]

J. L. Anderson and P. G. Bergmann, Constraints in covariant field theories, Phys. Rev., 83 (1951), 1018-1025.  doi: 10.1103/PhysRev.83.1018.  Google Scholar

[3]

A. Awane, $k$-symplectic structures, J. Math. Phys., 33 (1992), 4046-4052.  doi: 10.1063/1.529855.  Google Scholar

[4]

C. Batlle, J. Gomis, J. Pons and N. Román-Roy, Equivalence between the Lagrangian and Hamiltonian formalism for constrained systems, J. Math. Phys., 27 (1986), 2953-2962.  doi: 10.1063/1.527274.  Google Scholar

[5]

L. Búa, I. Bucataru, M. de León, M. Salgado and S. Vilariño, Symmetries in Lagrangian field theory, Rep. Math. Phys., 75 (2015), 333-357.  doi: 10.1016/S0034-4877(15)30010-0.  Google Scholar

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D. Chinea, M. de León and J. C. Marrero, Locally conformal cosymplectic manifolds and time-dependent Hamiltonian systems, Comment. Math. Univ. Carolin., 32 (1991), 383-387.   Google Scholar

[7]

D. Chinea, M. de León and J. C. Marrero, The constraint algorithm for time-dependent Lagrangians, J. Math. Phys., 35 (1994), 3410-3447.  doi: 10.1063/1.530476.  Google Scholar

[8]

P. Dirac, Generalized Hamiltonian dynamics, Can. J. Math., 2 (1950), 129-148.  doi: 10.4153/CJM-1950-012-1.  Google Scholar

[9]

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

[10]

M. J. Gotay and J. M. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence theorem, Ann. Inst. Henri Poincaré, 30 (1979), 129-142.   Google Scholar

[11]

M. J. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys., 19 (1978), 2388-2399.  doi: 10.1063/1.523597.  Google Scholar

[12]

X. Gràcia and J. M. Pons, A generalized geometric framework for constrained systems, Diff. Geom. Appl., 2 (1992), 223-247.  doi: 10.1016/0926-2245(92)90012-C.  Google Scholar

[13]

X. Gràcia and R. Martín, Geometric aspects of time-dependent singular differential equations, Int. J. Geom. Methods Mod. Phys., 2 (2005), 597-618.  doi: 10.1142/S0219887805000697.  Google Scholar

[14]

X. Gràcia, R. Martín and N. Román-Roy, Constraint algorithm for $k$-presymplectic Hamiltonian systems: Application to singular field theories, Int. J. Geom. Methods Mod. Phys., 6 (2009), 851-872.  doi: 10.1142/S0219887809003795.  Google Scholar

[15]

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

[16]

L. A. Ibort and J. Marín-Solano, A geometric classification of Lagrangian functions and the reduction of evolution space, J. Phys. A: Math. Gen., 25 (1992), 3353-3367.  doi: 10.1088/0305-4470/25/11/036.  Google Scholar

[17]

M. de León, J. Marín-Solano, and J. C. Marrero, A geometrical approach to classical field theories: A constraint algorithm for singular theories, In New Developments in Differential Geometry, Springer, Netherlands, 350 (1996), 291–312. doi: 10.1007/978-94-009-0149-0_22.  Google Scholar

[18]

M. de León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz-Lecanda and N. Román-Roy, Singular Lagrangian systems on jet bundles, Fortschr. Phys., 50 (2002), 105-169.  doi: 10.1002/1521-3978(200203)50:2<105::AID-PROP105>3.0.CO;2-N.  Google Scholar

[19]

M. de León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz-Lecanda and N. Román-Roy, Pre-multisymplectic constraint algorithm for field theories, Int. J. Geom. Meth. Mod. Phys., 2 (2005), 839-871.  doi: 10.1142/S0219887805000880.  Google Scholar

[20]

M. de León, E. Merino, J. A. Oubiña, P. 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

[21]

M. de León, E. 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

[22]

M. de León, M. Salgado and S. Vilariño, Methods of Differential Geometry in Classical Field Theories: $k$-Symplectic and $k$-Cosymplectic Approaches, World Scientific, Hackensack, 2016. doi: 10.1142/9693.  Google Scholar

[23]

G. Marmo, G. Mendella and W. M. Tulczyjew, Constrained Hamiltonian systems as implicit differential equations, J. Phys. A, 30 (1997), 277-293.  doi: 10.1088/0305-4470/30/1/020.  Google Scholar

[24]

M. C. Muñoz-Lecanda and N. Román-Roy, Lagrangian theory for presymplectic systems, Ann. Inst. Henry Poincaré: Phys. Theor., 57 (1992), 27-45.   Google Scholar

[25]

A. M. Rey, N. Román-Roy, M. Salgado and S. Vilariño, $k$-cosymplectic classical field theories: Tulckzyjew and Skinner–Rusk formulations, Math. Phys. Anal. Geom., 15 (2012), 85-119.  doi: 10.1007/s11040-012-9104-z.  Google Scholar

[26]

A. M. Rey, N. Román-Roy, M. Salgado and S. Vilariño, On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories, J. Geom. Mechs., 3 (2011), 113-137.  doi: 10.3934/jgm.2011.3.113.  Google Scholar

[27]

N. Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories, Symmetry Integrability Geom. Methods Appl (SIGMA), 5 (2009), Paper 100, 25 pp. doi: 10.3842/SIGMA.2009.100.  Google Scholar

[28]

E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective, Wiley, New York, 1974. doi: 10.1142/9751.  Google Scholar

[29]

K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 169, Springer, Berlin, 1982. doi: 10.1007/BFb0036225.  Google Scholar

[30]

S. Vignolo, A new presymplectic framework for time-dependent Lagrangian systems: the constraint algorithm and the second-order differential equation problem, J. Phys. A: Math. Gen., 33 (2000), 5117-5135.  doi: 10.1088/0305-4470/33/28/314.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Addison-Wesley, California, 2nd edition, 1978. doi: 10.1090/chel/364.  Google Scholar

[2]

J. L. Anderson and P. G. Bergmann, Constraints in covariant field theories, Phys. Rev., 83 (1951), 1018-1025.  doi: 10.1103/PhysRev.83.1018.  Google Scholar

[3]

A. Awane, $k$-symplectic structures, J. Math. Phys., 33 (1992), 4046-4052.  doi: 10.1063/1.529855.  Google Scholar

[4]

C. Batlle, J. Gomis, J. Pons and N. Román-Roy, Equivalence between the Lagrangian and Hamiltonian formalism for constrained systems, J. Math. Phys., 27 (1986), 2953-2962.  doi: 10.1063/1.527274.  Google Scholar

[5]

L. Búa, I. Bucataru, M. de León, M. Salgado and S. Vilariño, Symmetries in Lagrangian field theory, Rep. Math. Phys., 75 (2015), 333-357.  doi: 10.1016/S0034-4877(15)30010-0.  Google Scholar

[6]

D. Chinea, M. de León and J. C. Marrero, Locally conformal cosymplectic manifolds and time-dependent Hamiltonian systems, Comment. Math. Univ. Carolin., 32 (1991), 383-387.   Google Scholar

[7]

D. Chinea, M. de León and J. C. Marrero, The constraint algorithm for time-dependent Lagrangians, J. Math. Phys., 35 (1994), 3410-3447.  doi: 10.1063/1.530476.  Google Scholar

[8]

P. Dirac, Generalized Hamiltonian dynamics, Can. J. Math., 2 (1950), 129-148.  doi: 10.4153/CJM-1950-012-1.  Google Scholar

[9]

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

[10]

M. J. Gotay and J. M. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence theorem, Ann. Inst. Henri Poincaré, 30 (1979), 129-142.   Google Scholar

[11]

M. J. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys., 19 (1978), 2388-2399.  doi: 10.1063/1.523597.  Google Scholar

[12]

X. Gràcia and J. M. Pons, A generalized geometric framework for constrained systems, Diff. Geom. Appl., 2 (1992), 223-247.  doi: 10.1016/0926-2245(92)90012-C.  Google Scholar

[13]

X. Gràcia and R. Martín, Geometric aspects of time-dependent singular differential equations, Int. J. Geom. Methods Mod. Phys., 2 (2005), 597-618.  doi: 10.1142/S0219887805000697.  Google Scholar

[14]

X. Gràcia, R. Martín and N. Román-Roy, Constraint algorithm for $k$-presymplectic Hamiltonian systems: Application to singular field theories, Int. J. Geom. Methods Mod. Phys., 6 (2009), 851-872.  doi: 10.1142/S0219887809003795.  Google Scholar

[15]

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

[16]

L. A. Ibort and J. Marín-Solano, A geometric classification of Lagrangian functions and the reduction of evolution space, J. Phys. A: Math. Gen., 25 (1992), 3353-3367.  doi: 10.1088/0305-4470/25/11/036.  Google Scholar

[17]

M. de León, J. Marín-Solano, and J. C. Marrero, A geometrical approach to classical field theories: A constraint algorithm for singular theories, In New Developments in Differential Geometry, Springer, Netherlands, 350 (1996), 291–312. doi: 10.1007/978-94-009-0149-0_22.  Google Scholar

[18]

M. de León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz-Lecanda and N. Román-Roy, Singular Lagrangian systems on jet bundles, Fortschr. Phys., 50 (2002), 105-169.  doi: 10.1002/1521-3978(200203)50:2<105::AID-PROP105>3.0.CO;2-N.  Google Scholar

[19]

M. de León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz-Lecanda and N. Román-Roy, Pre-multisymplectic constraint algorithm for field theories, Int. J. Geom. Meth. Mod. Phys., 2 (2005), 839-871.  doi: 10.1142/S0219887805000880.  Google Scholar

[20]

M. de León, E. Merino, J. A. Oubiña, P. 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

[21]

M. de León, E. 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

[22]

M. de León, M. Salgado and S. Vilariño, Methods of Differential Geometry in Classical Field Theories: $k$-Symplectic and $k$-Cosymplectic Approaches, World Scientific, Hackensack, 2016. doi: 10.1142/9693.  Google Scholar

[23]

G. Marmo, G. Mendella and W. M. Tulczyjew, Constrained Hamiltonian systems as implicit differential equations, J. Phys. A, 30 (1997), 277-293.  doi: 10.1088/0305-4470/30/1/020.  Google Scholar

[24]

M. C. Muñoz-Lecanda and N. Román-Roy, Lagrangian theory for presymplectic systems, Ann. Inst. Henry Poincaré: Phys. Theor., 57 (1992), 27-45.   Google Scholar

[25]

A. M. Rey, N. Román-Roy, M. Salgado and S. Vilariño, $k$-cosymplectic classical field theories: Tulckzyjew and Skinner–Rusk formulations, Math. Phys. Anal. Geom., 15 (2012), 85-119.  doi: 10.1007/s11040-012-9104-z.  Google Scholar

[26]

A. M. Rey, N. Román-Roy, M. Salgado and S. Vilariño, On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories, J. Geom. Mechs., 3 (2011), 113-137.  doi: 10.3934/jgm.2011.3.113.  Google Scholar

[27]

N. Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories, Symmetry Integrability Geom. Methods Appl (SIGMA), 5 (2009), Paper 100, 25 pp. doi: 10.3842/SIGMA.2009.100.  Google Scholar

[28]

E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective, Wiley, New York, 1974. doi: 10.1142/9751.  Google Scholar

[29]

K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 169, Springer, Berlin, 1982. doi: 10.1007/BFb0036225.  Google Scholar

[30]

S. Vignolo, A new presymplectic framework for time-dependent Lagrangian systems: the constraint algorithm and the second-order differential equation problem, J. Phys. A: Math. Gen., 33 (2000), 5117-5135.  doi: 10.1088/0305-4470/33/28/314.  Google Scholar

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