Publication:
Poisson–Poincaré reduction for Field Theories

Loading...
Thumbnail Image
Official URL
Full text at PDC
Publication Date
2022-10-27
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Citations
Google Scholar
Research Projects
Organizational Units
Journal Issue
Abstract
Given a Hamiltonian system on a fiber bundle, there is a Poisson covariant formulation of the Hamilton equations. When a Lie group G acts freely, properly, preserving the fibers of the bundle and the Hamiltonian density is G-invariant, we study the reduction of this formulation to obtain an analogue of Poisson–Poincaré reduction for field theories. This procedure is related to the Lagrange–Poincaré reduction for field theories via a Legendre transformation. Finally, an application to a model of a charged strand evolving in an electric field is given.
Description
Unesco subjects
Keywords
Citation
[1] M. A. Berbel and M. Castrillón López. Reduction (by stages) in the whole Lagrange–Poincaré category. arXiv:1912.10763, 2019. [2] M. A. Berbel and M. Castrillón López. Lagrangian reduction by stages in field theory. arXiv: 2007.14854, 2020. [3] C. Blacker. Reduction of multisymplectic manifolds. Lett. Math. Phys., 111(3):Paper No. 64, 30, 2021. [4] H. Bursztyn, A. Cabrera, and D. Iglesias. Multisymplectic geometry and Lie groupoids. In Geometry, mechanics, and dynamics, volume 73 of Fields Inst. Commun., pages 57–73. Springer, New York, 2015. [5] S. Capriotti, V. A. Daz, E. G.-T. Andrs, and T. Mestdag. Cotangent bundle reduction and routh reduction for polysymplectic manifolds, 2022. [6] J. F. Cariñena, M. Crampin, and L. A. Ibort. On the multisymplectic formalism for first order field theories. Differential Geom. Appl., 1(4):345–374, 1991. [7] 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., 74:352–369, 2013. [8] M. Castrillón López, P. L. García Pérez, and T. S. Ratiu. Euler-Poincaré reduction on principal bundles. Lett. Math. Phys., 58(2):167–180, 2001. [9] M. Castrillón López and J. E. Marsden. Some remarks on Lagrangian and Poisson reduction for field theories. J. Geom. Phys., 48(1):52–83, 2003. [10] M. Castrillón López and T. S. Ratiu. Reduction in principal bundles: covariant Lagrange-Poincar´e equations. Comm. Math. Phys., 236(2):223–250, 2003. [11] H. Cendra, J. E. Marsden, and T. S. Ratiu. Lagrangian reduction by stages. Mem. Amer. Math. Soc., 152(722):1–108, 2001. [12] F. M. Ciaglia, F. Di Cosmo, A. Ibort, L. Marmo, G. Schiavone, and A. Zampini. The geometry of the solution space of first order hamiltonian field theories I: from particle dynamics to free electrodynamics. arXiv: 2208.14136, 2022. [13] F. M. Ciaglia, F. Di Cosmo, A. Ibort, L. Marmo, G. Schiavone, and A. Zampini. The geometry of the solution space of first order hamiltonian field theories II: non-abelian gauge theories. arXiv: 2208.14155, 2022. [14] J. de Lucas, X. Gràcia, X. Rivas, N. Román-Roy, and S. Vilariño. Reduction and reconstruction of multisymplectic Lie systems. J. Phys. A, 55(29):Paper No. 295204, 34, 2022. 32 [15] A. Echeverría-Enríquez, M. C. Muñoz Lecanda, and N. Román-Roy. Remarks on multisymplectic reduction. Rep. Math. Phys., 81(3):415–424, 2018. [16] 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(3):811–902, 2010. [17] D. C. P. Ellis, F. Gay-Balmaz, D. D. Holm, and T. S. Ratiu. Lagrange-Poincaré field equations. J. Geom. Phys., 61(11):2120–2146, 2011. [18] M. J. Gotay, J. Isenberg, J. E. Marsden, and R. Montgomery. Momentum maps and classical relativistic fields. part I: Covariant field theory. arXiv: 9801019, 2004. [19] C. Günther. The polysymplectic Hamiltonian formalism in field theory and calculus of variations. I. The local case. J. Differential Geom., 25(1):23–53, 1987. [20] D. D. Holm and R. I. Ivanov. Matrix G-strands. Nonlinearity, 27(6):1445–1469, 2014. [21] D. D. Holm, R. I. Ivanov, and J. R. Percival. G-strands. J. Nonlinear Sci., 22(4):517–551, 2012. [22] I. V. Kanatchikov. Canonical structure of classical field theory in the polymomentum phase space. Rep. Math. Phys., 41(1):49–90, 1998. [23] T. B. Madsen and A. Swann. Multi-moment maps. Adv. Math., 229(4):2287–2309, 2012. [24] T. B. Madsen and A. Swann. Closed forms and multi-moment maps. Geom. Dedicata, 165:25–52, 2013. [25] J. C. Marrero, N. Román-Roy, M. Salgado, and S. Vilariño. Reduction of polysymplectic manifolds. J. Phys. A, 48(5):055206, 43, 2015. [26] J. Marsden and A. Weinstein. Reduction of symplectic manifolds with symmetry. Rep. Mathematical Phys., 5(1):121–130, 1974. [27] J. E. Marsden, G. Misiolek, J.-P. Ortega, M. Perlmutter, and T. S. Ratiu. Hamiltonian Reduction by Stages, volume 1913 of Lecture Notes in Mathematics. Springer, Berlin, Heidelberg, 1 edition, 2007. [28] J. E. Marsden, R. Montgomery, P. J. Morrison, and W. B. Thompson. Covariant Poisson brackets for classical fields. Ann. Physics, 169(1):29–47, 1986. [29] J. E. Marsden and T. Ratiu. Reduction of Poisson manifolds. Lett. Math. Phys., 11(2):161–169, 1986. [30] J. E. Marsden and T. S. Ratiu. Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, volume 17 of Texts in Applied Mathematics. Springer, New York, NY, 2 edition, 1999. [31] J. E. Marsden and A. Weinstein. Comments on the history, theory, and applications of symplectic reduction. In Quantization of singular symplectic quotients, volume 198 of Progr. Math., pages 1–19. Birkhäuser, Basel, 2001. [32] J.-P. Ortega and T. S. Ratiu. Momentum maps and Hamiltonian reduction, volume 222 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, 2004. [33] J. M. Souriau. Structure of dynamical systems, volume 149 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, 1997. [34] K. Uhlenbeck. Harmonic maps into Lie groups: classical solutions of the chiral model. J. Differential Geom., 30(1):1–50, 1989. [35] J. Vankerschaver, H. Yoshimura, and M. Leok. The Hamilton-Pontryagin principle and multi-Dirac structures for classical field theories. J. Math. Phys., 53(7):072903, 25, 2012. [36] H. Yoshimura and J. E. Marsden. Dirac cotangent bundle reduction. J. Geom. Mech., 1(1):87–158, 2009.
Collections