# American Institute of Mathematical Sciences

March  2014, 6(1): 1-23. doi: 10.3934/jgm.2014.6.1

## Aspects of reduction and transformation of Lagrangian systems with symmetry

 1 Department of Mathematics, Ghent University, Krijgslaan 281 S22, B9000 Ghent, Belgium, Belgium 2 Belgian Institute for Space Aeronomy, Ringlaan 3, B1180 Brussels, Belgium

Received  July 2013 Revised  January 2014 Published  April 2014

This paper contains results on geometric Routh reduction and it is a continuation of a previous paper [7] where a new class of transformations is introduced between Lagrangian systems obtained after Routh reduction. In general, these reduced Lagrangian systems have magnetic force terms and are singular in the sense that the Lagrangian does not depend on some velocity components. The main purpose of this paper is to show that the Routh reduction process itself is entirely captured by the application of such a new transformation on the initial Lagrangian system with symmetry.
Citation: E. García-Toraño Andrés, Bavo Langerock, Frans Cantrijn. Aspects of reduction and transformation of Lagrangian systems with symmetry. Journal of Geometric Mechanics, 2014, 6 (1) : 1-23. doi: 10.3934/jgm.2014.6.1
##### References:

show all references

##### References:
 [1] Katarzyna Grabowska, Paweƚ Urbański. Geometry of Routh reduction. Journal of Geometric Mechanics, 2019, 11 (1) : 23-44. doi: 10.3934/jgm.2019002 [2] Leonardo J. Colombo, María Emma Eyrea Irazú, Eduardo García-Toraño Andrés. A note on Hybrid Routh reduction for time-dependent Lagrangian systems. Journal of Geometric Mechanics, 2020, 12 (2) : 309-321. doi: 10.3934/jgm.2020014 [3] Joshua Cape, Hans-Christian Herbig, Christopher Seaton. Symplectic reduction at zero angular momentum. Journal of Geometric Mechanics, 2016, 8 (1) : 13-34. doi: 10.3934/jgm.2016.8.13 [4] Sergio Grillo, Marcela Zuccalli. Variational reduction of Lagrangian systems with general constraints. Journal of Geometric Mechanics, 2012, 4 (1) : 49-88. doi: 10.3934/jgm.2012.4.49 [5] Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of discrete mechanical systems by stages. Journal of Geometric Mechanics, 2016, 8 (1) : 35-70. doi: 10.3934/jgm.2016.8.35 [6] Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems. Journal of Geometric Mechanics, 2010, 2 (1) : 69-111. doi: 10.3934/jgm.2010.2.69 [7] C. D. Ahlbrandt, A. C. Peterson. A general reduction of order theorem for discrete linear symplectic systems. Conference Publications, 1998, 1998 (Special) : 7-18. doi: 10.3934/proc.1998.1998.7 [8] L. Búa, T. Mestdag, M. Salgado. Symmetry reduction, integrability and reconstruction in $k$-symplectic field theory. Journal of Geometric Mechanics, 2015, 7 (4) : 395-429. doi: 10.3934/jgm.2015.7.395 [9] Hiroaki Yoshimura, Jerrold E. Marsden. Dirac cotangent bundle reduction. Journal of Geometric Mechanics, 2009, 1 (1) : 87-158. doi: 10.3934/jgm.2009.1.87 [10] Inês Cruz, M. Esmeralda Sousa-Dias. Reduction of cluster iteration maps. Journal of Geometric Mechanics, 2014, 6 (3) : 297-318. doi: 10.3934/jgm.2014.6.297 [11] Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1 [12] Jean-François Babadjian, Francesca Prinari, Elvira Zappale. Dimensional reduction for supremal functionals. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1503-1535. doi: 10.3934/dcds.2012.32.1503 [13] Thomas Espitau, Antoine Joux. Certified lattice reduction. Advances in Mathematics of Communications, 2020, 14 (1) : 137-159. doi: 10.3934/amc.2020011 [14] Martins Bruveris, David C. P. Ellis, Darryl D. Holm, François Gay-Balmaz. Un-reduction. Journal of Geometric Mechanics, 2011, 3 (4) : 363-387. doi: 10.3934/jgm.2011.3.363 [15] Ning Zhang, Qiang Wu. Online learning for supervised dimension reduction. Mathematical Foundations of Computing, 2019, 2 (2) : 95-106. doi: 10.3934/mfc.2019008 [16] Andrei Korobeinikov, Aleksei Archibasov, Vladimir Sobolev. Order reduction for an RNA virus evolution model. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1007-1016. doi: 10.3934/mbe.2015.12.1007 [17] Andrejs Reinfelds, Klara Janglajew. Reduction principle in the theory of stability of difference equations. Conference Publications, 2007, 2007 (Special) : 864-874. doi: 10.3934/proc.2007.2007.864 [18] Rentsen Enkhbat, Evgeniya A. Finkelstein, Anton S. Anikin, Alexandr Yu. Gornov. Global optimization reduction of generalized Malfatti's problem. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 211-221. doi: 10.3934/naco.2017015 [19] Marco Castrillón López, Pablo M. Chacón, Pedro L. García. Lagrange-Poincaré reduction in affine principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 399-414. doi: 10.3934/jgm.2013.5.399 [20] Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201

2019 Impact Factor: 0.649