Aspects of reduction and transformation of Lagrangian systems with symmetry

E. García-Toraño Andrés; Bavo Langerock; Frans Cantrijn

doi:10.3934/jgm.2014.6.1

Article Contents

2014, Volume 6, Issue 1: 1-23. Doi: 10.3934/jgm.2014.6.1

This issue Previous Article Next Article Andoyer's variables and phases in the free rigid body

Aspects of reduction and transformation of Lagrangian systems with symmetry

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

Received: June 30, 2013

Revised: December 31, 2013

Published: February 2014

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 37J05, 37J15, 53D20.

Citation:

Related Papers

Cited by

References

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics, The Benjamin/Cummings Publishing Company, INC, 1978.
[2]	M. Crampin and T. Mestdag, Routh's procedure for non-Abelian symmetry groups, J. Math. Phys., 49 (2008), 032901.doi: 10.1063/1.2885077.
[3]	M. J. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys., 19 (1978), 2388-2399.doi: 10.1063/1.523597.
[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), 129-142.
[5]	S. M. Jalnapurkar and J. E. Marsden, Reduction of Hamilton's variational principle, Dynamics and Stability of Systems, 15 (2000), 287-318.doi: 10.1080/713603744.
[6]	S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, volume I and II, Interscience Publishers, 1963.
[7]	B. Langerock, E. G. Andrés and F. Cantrijn, Routh reduction and the class of magnetic Lagrangian systems, Journal Of Mathematical Physics, 53 (2012), 062902, 19 pp.doi: 10.1063/1.4723841.
[8]	B. Langerock, F. Cantrijn and J. Vankerschaver, Routhian reduction for quasi-invariant Lagrangians, J. Math. Phys., 51 (2010), 022902.doi: 10.1063/1.3277181.
[9]	B. Langerock and M. C. Lopéz, Routhian reduction for singular Lagrangians, J. Geom. Meth. Mod. Phys., 7 (2010), 1451-1489.doi: 10.1142/S0219887810004907.
[10]	B. Langerock, T. Mestdag and J. Vankerschaver, Routh reduction by stages, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), 31pp.doi: 10.3842/SIGMA.2011.109.
[11]	J. E. Marsden, Lectures on Mechanics, Cambridge University Press, 1992.
[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, Berlin, 2007.
[13]	J. E. Marsden, T. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations, J. Math. Phys., 41 (2000), 3379-3429.doi: 10.1063/1.533317.
[14]	J. E. Marsden and J. Scheurle, Lagrangian reduction and the double spherical pendulum, Z. Angew. Math. Phys., 44 (1993), 17-43.doi: 10.1007/BF00914351.
[15]	P. Morando and S. Sammarco, Variational problems with symmetries: A Pfaffian system approach, Acta Appl. Math., 120 (2012), 255-274.doi: 10.1007/s10440-012-9720-4.
[16]	R. W. Sharpe, Differential Geometry, volume 166 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. Cartan's generalization of Klein's Erlangen program, With a foreword by S. S. Chern.