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In memory of Geneviève Raugel
Constraint algorithm for singular field theories in the k-cosymplectic framework
Department of Mathematics, Universitat Politècnica de Catalunya, Campus Nord UPC, edifici C3, C. Jordi Girona, 1, 08034 Barcelona, Catalonia, Spain |
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.
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics, Addison-Wesley, California, 2nd edition, 1978.
doi: 10.1090/chel/364. |
[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. |
[3] |
A. Awane,
$k$-symplectic structures, J. Math. Phys., 33 (1992), 4046-4052.
doi: 10.1063/1.529855. |
[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. |
[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. |
[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.
|
[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. |
[8] |
P. Dirac,
Generalized Hamiltonian dynamics, Can. J. Math., 2 (1950), 129-148.
doi: 10.4153/CJM-1950-012-1. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
M. C. Muñoz-Lecanda and N. Román-Roy,
Lagrangian theory for presymplectic systems, Ann. Inst. Henry Poincaré: Phys. Theor., 57 (1992), 27-45.
|
[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. |
[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. |
[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. |
[28] |
E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective, Wiley, New York, 1974.
doi: 10.1142/9751. |
[29] |
K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 169, Springer, Berlin, 1982.
doi: 10.1007/BFb0036225. |
[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. |
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. |
[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. |
[3] |
A. Awane,
$k$-symplectic structures, J. Math. Phys., 33 (1992), 4046-4052.
doi: 10.1063/1.529855. |
[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. |
[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. |
[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.
|
[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. |
[8] |
P. Dirac,
Generalized Hamiltonian dynamics, Can. J. Math., 2 (1950), 129-148.
doi: 10.4153/CJM-1950-012-1. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
M. C. Muñoz-Lecanda and N. Román-Roy,
Lagrangian theory for presymplectic systems, Ann. Inst. Henry Poincaré: Phys. Theor., 57 (1992), 27-45.
|
[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. |
[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. |
[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. |
[28] |
E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective, Wiley, New York, 1974.
doi: 10.1142/9751. |
[29] |
K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 169, Springer, Berlin, 1982.
doi: 10.1007/BFb0036225. |
[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. |
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