Lagrangian reduction by stages in field theory

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We propose a category of bundles in order to perform Lagrangian reduction by stages in covariant Field Theory. This category plays an analogous role to Lagrange-Poincaré bundles in Lagrangian reduction by stages in Mechanics and includes both jet bundles and reduced covariant configuration spaces. Furthermore, we analyze the resulting reconstruction condition and formulate the Noether theorem in this context. Finally, a model of a molecular strand with rotors is seen as an application of this theoretical frame.
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[1] R. Abraham, J.E. Marsden, Foundations of Mechanics. Benjamin/Cummings Publishing, Advanced Book Program, Reading (1978). [2] V.I. Arnold, Mathematical Methods of Classical Mechanics, 2nd edn. Graduate Texts in Mathematics, vol. 60. Springer, New York (1989) [3] M.A. Berbel, M. Castrillón López, Reduction (by stages) in the whole Lagrange–Poincaré category, arXiv:1912.10763 [4] A. Bloch, L. Colombo, F. Jiménez, The variational discretization of the constrained higher-order Lagrange-Poincaré equations, Discrete Contin. Dyn. Syst. 39 (2019), no. 1, 309–344. [5] M. Castrillón López, P.L. García, C. Rodrigo, EulerPoincaré reduction in principal bundles by a subgroup of the structure group J. Geom. Phys. 74 (2013) 352369. [6] M. Castrillón López, P.L. García Pérez, The problem of Lagrange on principal bundles under a subgroup of symmetries J. Geom. Mech. 11, No. 4, (2019) 539–552 . [7] M. Castrillón López, P. L. García Pérez, T.S. Ratiu, Euler-Poincaré Reduction on Principal Bundles. Lett. Math. Phys 58 (2001), 167–180. [8] M. Castrillón López, T. S. Ratiu, Reduction in Principal Bundles: Covariant Lagrange-Poincaré Equations Commun. Math. Phys. 236, 223250 (2003) [9] H. Cendra, J.E. Marsden, T.S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc. 152 (2001), no. 722. [10] F. Crespo, S. Ferrer, J.C. van der Meer, SO(3) × T 4 -reduction and relative equilibria for a radial axisymmetric intermediary model for roto-orbital motion, J. Geom. Phys. 150, (2020). [11] D. Ellis, F. Gay-Balmaz, D. Holm, V. Putzaradze, T. Ratiu, Symmetry Reduced Dynamics of Charged Molecular Strands Arch. Rational Mech. Anal. 197 (2010) 811–902. [12] D. Ellis, F. Gay-Balmaz, D. Holm, T.S. Ratiu, Lagrange–Poincaré field equations Journal of Geometry and Physics, 61(11) (2011), 2120–2146. [13] J. Fern´andez, C. Tori, M. Zuccalli, Lagrangian reduction of discrete mechanical systems by stages, J. Geom. Mech. 8 (2016), no. 1, 35–70. [14] F. Gay-Balmaz, M. Monastyrsky, T.S. Ratiu, Lagrangian reductions and integrable systems in condensed matter, Comm. Math. Phys. 335 (2015), no. 2, 609–636. [15] M. Gotay, J. Isenberg, J.E. Marsden, R. Montgomery, Momentum Maps and Classical Fields. Part I: Covariant Field Theory, arXiv:physics/9801019 [16] S.D. Grillo, L.M. Salomone, M. Zuccalli, Variational reduction of Hamiltonian systems with general constraints, J. Geom. Phys. 144 (2019), 209–234. [17] J.E. Marsden, J. Scheurle, The reduced Euler-Lagrange equations. Fields Institute Comm, 1 139–164, 1993. [18] J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, 2nd edn. Texts in Applied Mathematics, vol. 17. Springer, New York (1999). [19] J. Janyska, M. Modugno, Relations between Linear Connections on the Tangent Bundle and Connections on the Jet Bundle of a Fiber Manifold Archivum Mathematicum Brno 32 (1996), 281–288. [20] J.K, Moser, C.L. Siegel, Lectures on celestial mechanics, Transl. from the German by C. I. Kalme. Reprint of the 1971 ed. (English) Classics in Mathematics. Berlin: Springer-Verlag. xii, 290 p. (1995).