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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

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, 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.

show all references

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.

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