Aspects of reduction and transformation of Lagrangian systems with symmetry

Previous Article
JGM Home
This Issue
Next Article
Andoyer's variables and phases in the free rigid body

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

Aspects of reduction and transformation of Lagrangian systems with symmetry

E. García-Toraño Andrés ^1, , Bavo Langerock ^2, and Frans Cantrijn ^1,

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

Abstract
Related Papers

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.

Keywords: Lagrangian reduction., Routh reduction, Symplectic reduction.

Mathematics Subject Classification: 37J05, 37J15, 53D2.

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:

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics,, The Benjamin/Cummings Publishing Company, (1978). Google Scholar
[2]	M. Crampin and T. Mestdag, Routh's procedure for non-Abelian symmetry groups,, J. Math. Phys., 49 (2008). doi: 10.1063/1.2885077. Google Scholar
[3]	M. J. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints,, J. Math. Phys., 19 (1978), 2388. doi: 10.1063/1.523597. Google Scholar
[4]	M. J. Gotay and J. M. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence problem,, Ann. Inst. Henri Poincaré, 30 (1979), 129. Google Scholar
[5]	S. M. Jalnapurkar and J. E. Marsden, Reduction of Hamilton's variational principle,, Dynamics and Stability of Systems, 15 (2000), 287. doi: 10.1080/713603744. Google Scholar
[6]	S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, volume I and II,, Interscience Publishers, (1963). Google Scholar
[7]	B. Langerock, E. G. Andrés and F. Cantrijn, Routh reduction and the class of magnetic Lagrangian systems,, Journal Of Mathematical Physics, 53 (2012). doi: 10.1063/1.4723841. Google Scholar
[8]	B. Langerock, F. Cantrijn and J. Vankerschaver, Routhian reduction for quasi-invariant Lagrangians,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3277181. Google Scholar
[9]	B. Langerock and M. C. Lopéz, Routhian reduction for singular Lagrangians,, J. Geom. Meth. Mod. Phys., 7 (2010), 1451. doi: 10.1142/S0219887810004907. Google Scholar
[10]	B. Langerock, T. Mestdag and J. Vankerschaver, Routh reduction by stages,, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011). doi: 10.3842/SIGMA.2011.109. Google Scholar
[11]	J. E. Marsden, Lectures on Mechanics,, Cambridge University Press, (1992). Google Scholar
[12]	J. E. Marsden, G. Misiołek, J. P. Ortega, M. Perlmutter and T. S. Ratiu, Hamiltonian Reduction by Stages,, volume 1913 of Lecture Notes in Mathematics. Springer, (1913). Google Scholar
[13]	J. E. Marsden, T. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations,, J. Math. Phys., 41 (2000), 3379. doi: 10.1063/1.533317. Google Scholar
[14]	J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum,, Z. Angew. Math. Phys., 44 (1993), 17. doi: 10.1007/BF00914351. Google Scholar
[15]	P. Morando and S. Sammarco, Variational problems with symmetries: A Pfaffian system approach,, Acta Appl. Math., 120 (2012), 255. doi: 10.1007/s10440-012-9720-4. Google Scholar
[16]	R. W. Sharpe, Differential Geometry,, volume 166 of Graduate Texts in Mathematics. Springer-Verlag, (1997). Google Scholar

show all references

References:

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics,, The Benjamin/Cummings Publishing Company, (1978). Google Scholar
[2]	M. Crampin and T. Mestdag, Routh's procedure for non-Abelian symmetry groups,, J. Math. Phys., 49 (2008). doi: 10.1063/1.2885077. Google Scholar
[3]	M. J. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints,, J. Math. Phys., 19 (1978), 2388. doi: 10.1063/1.523597. Google Scholar
[4]	M. J. Gotay and J. M. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence problem,, Ann. Inst. Henri Poincaré, 30 (1979), 129. Google Scholar
[5]	S. M. Jalnapurkar and J. E. Marsden, Reduction of Hamilton's variational principle,, Dynamics and Stability of Systems, 15 (2000), 287. doi: 10.1080/713603744. Google Scholar
[6]	S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, volume I and II,, Interscience Publishers, (1963). Google Scholar
[7]	B. Langerock, E. G. Andrés and F. Cantrijn, Routh reduction and the class of magnetic Lagrangian systems,, Journal Of Mathematical Physics, 53 (2012). doi: 10.1063/1.4723841. Google Scholar
[8]	B. Langerock, F. Cantrijn and J. Vankerschaver, Routhian reduction for quasi-invariant Lagrangians,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3277181. Google Scholar
[9]	B. Langerock and M. C. Lopéz, Routhian reduction for singular Lagrangians,, J. Geom. Meth. Mod. Phys., 7 (2010), 1451. doi: 10.1142/S0219887810004907. Google Scholar
[10]	B. Langerock, T. Mestdag and J. Vankerschaver, Routh reduction by stages,, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011). doi: 10.3842/SIGMA.2011.109. Google Scholar
[11]	J. E. Marsden, Lectures on Mechanics,, Cambridge University Press, (1992). Google Scholar
[12]	J. E. Marsden, G. Misiołek, J. P. Ortega, M. Perlmutter and T. S. Ratiu, Hamiltonian Reduction by Stages,, volume 1913 of Lecture Notes in Mathematics. Springer, (1913). Google Scholar
[13]	J. E. Marsden, T. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations,, J. Math. Phys., 41 (2000), 3379. doi: 10.1063/1.533317. Google Scholar
[14]	J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum,, Z. Angew. Math. Phys., 44 (1993), 17. doi: 10.1007/BF00914351. Google Scholar
[15]	P. Morando and S. Sammarco, Variational problems with symmetries: A Pfaffian system approach,, Acta Appl. Math., 120 (2012), 255. doi: 10.1007/s10440-012-9720-4. Google Scholar
[16]	R. W. Sharpe, Differential Geometry,, volume 166 of Graduate Texts in Mathematics. Springer-Verlag, (1997). Google Scholar

[1]	Katarzyna Grabowska, Paweƚ Urbański. Geometry of Routh reduction. Journal of Geometric Mechanics, 2019, 11 (1) : 23-44. doi: 10.3934/jgm.2019002
[2]	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
[3]	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
[4]	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
[5]	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
[6]	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
[7]	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
[8]	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
[9]	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
[10]	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
[11]	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
[12]	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
[13]	Ning Zhang, Qiang Wu. Online learning for supervised dimension reduction. Mathematical Foundations of Computing, 2019, 2 (2) : 95-106. doi: 10.3934/mfc.2019008
[14]	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
[15]	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
[16]	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
[17]	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
[18]	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
[19]	A.V. Borisov, A.A. Kilin, I.S. Mamaev. Reduction and chaotic behavior of point vortices on a plane and a sphere. Conference Publications, 2005, 2005 (Special) : 100-109. doi: 10.3934/proc.2005.2005.100
[20]	Henrique Bursztyn, Alejandro Cabrera. Symmetries and reduction of multiplicative 2-forms. Journal of Geometric Mechanics, 2012, 4 (2) : 111-127. doi: 10.3934/jgm.2012.4.111

2018 Impact Factor: 0.525

American Institute of Mathematical Sciences