Reduction of Forward Difference Operators in Principal G-bundles
Abstract
Retraction maps on Lie groups can be successfully used in mechanics and control theory to generate numerical integration schemes, for ordinary differential equations with a variational origin, recovering at the same time a discrete version of the energy and symplectic structure conservation properties, that are characteristic of smooth variational mechanics. The present work fixes the specific tool that plays in gauge field theories the same role as retraction maps on geometric mechanics. This tool, the covariant reduced projectable forward difference operator, can be used for a covariant discretization of the main elements of a variational theory: the jet bundle, the Lagrangian density and the associated action functional. Particular interest is dedicated to the trivialized formulation of a gauge field theory, and its reduction into a theory where fields are given as principal connections and $H$-structures. Main characteristics of the presented method are its covariance by gauge transformations and the commutation of the discretization and the reduction processes.References
P.A. Absil, R. Mahony, and R. Sepulchre. Optimization algorithms on matrix manifolds. Princeton Univ. Press, Princeton, NJ, 2008
R.L. Adler, J.P. Dedieu, J.Y. Margulies, M. Martens, M. Shub. Newton’s method on Riemannian manifolds and a geometric model for the human spine. IMA Journal of Numerical Analysis, 22(3), (2002) 359–390.
M. F. Atiyah, Complex analytic connections in fibre bundles. Transactions of the American Mathematical Society 85.1 (1957): 181–207.
I. Biswas, F. Neumann. Atiyah sequences, connections and characteristic forms for principal bundles over groupoids and stacks. Comptes Rendus Mathematique 352.1 (2014): 59–64.
A.I. Bobenko, Yu.B. Suris. Discrete Lagrangian reduction, discrete Euler–Poincaré equations, and semidirect products. Letters in Mathematical Physics 49.1 (1999): 79–93.
R.D. Bos. Continuous representations of groupoids. Houston J. Math 37.3 (2011): 807-844.
R.D. Bos. Groupoids in geometric quantization. PhD Thesis Radboud University of Nijmegen, 2007.
N. Bou-Rabee, J.E. Marsden. Hamilton–Pontryagin integrators on Lie groups part I: Introduction and structure-preserving properties Foundations of Computational Mathematics 9.2 (2009): 197–219.
A. Cannas da Silva, A. Weinstein. Geometric models for noncommutative algebras. Vol. 10. American Mathematical Soc., 1999.
A.C. Casimiro, C. Rodrigo, First variation formula and conservation laws in several independent discrete variables. J. Geom. Phys. 62.1 (2012), 61–86.
A.C. Casimiro, C. Rodrigo, First variation formula for discrete variational problems in two independent variables. RACSAM Rev. R. Acad. A 106.1 (2012), 111–135.
A.C. Casimiro, C. Rodrigo, Variational Integrators for reduced field equations, Statistics Opt. Inform. Comput., 6 (2018), 84-113.
M. Castrillón López, Field theories: reduction, constraints and variational integrators. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matematicas 106(1) (2012): 67–74
M. Castrillón López, P.L. García, T.S. Ratiu, Euler-Poincaré reduction on principal bundles, Lett. Math. Phys. 58 (2001) 167–180
M. Castrillón López, P.L. García, C. Rodrigo, Euler-Poincaré reduction in principal bundles by a subgroup of the structure group, Journal of Geometry and Physics, Volume 74, December 2013, Pages 352–369
K. Crane, Discrete connections for geometry processing. Diss. California Institute of Technology, 2010.
P.E. Crouch, R. Grossman. Numerical integration of ordinary differential equations on manifolds. Journal of Nonlinear Science 3.1
(1993): 1–33.
F. Demoures, F. Gay-Balmaz, S. Leyendecker, S. Ober-Bl?baum, T.S. Ratiu, T. S., Y. Weinand. Discrete variational Lie group formulation of geometrically exact beam dynamics. Numerische Mathematik, 130(1), (2015) 73–123.
F. Demoures, F. Gay-Balmaz, T.S. Ratiu. Multisymplectic variational integrators and space/time symplecticity. Analysis and Applications, 14(03), (2016) 341–391.
A. Edelman, T.A. Arias, S.T. Smith. The geometry of algorithms with orthogonality constraints. SIAM journal on Matrix Analysis and Applications 20.2 (1998): 303–353.
C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Coll. de Topologie, Bruxelles, CBRM (1950), 29—55.
D.C.P. Ellis, F. Gay-Balmaz, D.D. Holm, V. Putkaradze, T.S. Ratiu, Symmetry reduced dynamics of charged molecular strands, Arch. Rat. Mech. and Anal., 197(2), 811–902 (2010)
D.C.P. Ellis, F. Gay-Balmaz, D.D. Holm, T.S. Ratiu, Lagrange-Poincaré field equations, J. Geom. Phys. 61 (11) (2011) 2120-2146.
J. Fernandez, M. Zuccalli. A geometric approach to discrete connections on principal bundles. Journal of Geometric Mechanics. Dec2013, Vol. 5 Issue 4, p433-444. 12p.
E.S. Gawlik, P. Mullen, D. Pavlov, J.E. Marsden, M. Desbrun. Geometric, variational discretization of continuum theories. Physica D: Nonlinear Phenomena, 240(21), (2011) 1724–1760.
A. Iserles, H.Z. Munthe-Kaas, S. P. N?rsett, A. Zanna. Lie-group methods. Acta Numerica, (2005) 1-148.
F. Jiménez, M. Kobilarov, D. Martín de Diego. Discrete variational optimal control. J. of Nonlinear Sci. 23.3 (2013): 393-426.
M. B. Kobilarov, J. E. Marsden. Discrete geometric optimal control on Lie groups IEEE Transactions on Robotics 27.4 (2011): 641-655.
M. Kobilarov. Solvability of Geometric Integrators for Multi-body Systems. Multibody Dynamics. Springer International Publishing, 2014. 145–174.
A. Kock. Synthetic geometry of manifolds. Vol. 180. Cambridge University Press, 2010.
R. McLachlan, M. Perlmutter. Integrators for nonholonomic mechanical systems Journal of nonlinear science 16.4 (2006): 283–328.
M. Leok, J.E. Marsden, and A.D. Weinstein. A discrete theory of connections on principal bundles. arXiv preprint math/0508338 (2005).
J.E. Marsden, S. Pekarsky, S. Shkoller. Symmetry reduction of discrete Lagrangian mechanics on Lie groups. Journal of geometry and physics 36.1 (2000): 140–151.
J.E. Marsden, J. Scheurle. The reduced Euler-Lagrange equations. Fields Institute Comm 1 (1993): 139–164.
A. Saccon. Midpoint rule for variational integrators on Lie groups. International journal for numerical methods in engineering 78.11 (2009): 1345.
D.J. Saunders. The geometry of jet bundles. Vol. 142. Cambridge University Press, 1989.
J.P. Serre, Lie algebras and Lie groups, Lectures given at Harvard University, 1964, Benjamin, New York, 1965.
M. Shub. Some Remarks on Dynamical Systems and Numerical Analysis, in: Dynamical Systems and Partial Differential Equations, Proceedings of VII ELAM (L. Lara-Carrero and J. Lewowicz eds.), Equinoccio, Universidad Simon Bolivar, Caracas, (1986), 69–92
J. Vankerschaver. Euler-Poincaré reduction for discrete field theories. Journal of mathematical physics, 48(3), (2007) 032902.
J. Vankerschaver, F. Cantrijn. Discrete Lagrangian field theories on Lie groupoids. Journal of Geometry and Physics 57(2) (2007):665–689.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
