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December  2019, 11(4): 539-552. doi: 10.3934/jgm.2019026

The problem of Lagrange on principal bundles under a subgroup of symmetries

Marco Castrillón López 1, and  Pedro Luis García Pérez 2,

1. 

Dept. Álgebra, Geometría y Topología, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
2. 

Dept. Matemática, Universidad de Salamanca, 37008 Salamanca, Spain and Real Academia de Ciencias, 28004 Madrid, Spain

We dedicate this work to our friend Darryl D. Holm on the occasion of his 70th birthday.

Received  April 2018 Revised  May 2019 Published  November 2019

Fund Project: This work has been partially supported by MICINN (Spain) under projects MTM2015-63612-P and PGC2018-098321-B-I00, as well as Consejería de Educación, Junta de Castilla-León (Spain) under project SA090G19.

Given a Lagrangian density $ L{\bf{v}} $ defined on the $ 1 $-jet extension $ J^1P $ of a principal $ G $-bundle $ \pi \colon P\to M $ invariant under the action of a closed subgroup $ H\subset G $, its Euler-Poincaré reduction in $ J^1P/H = C(P)\times_M P/H $ ($ C(P)\to M $ being the bundle of connections of $ P $ and $ P/H\to M $ being the bundle of $ H $-structures) induces a Lagrange problem defined in $ J^1(C(P)\times_M P/H) $ by a reduced Lagrangian density $ l{\bf{v}} $ together with the constraints $ {\rm{Curv}}\sigma = 0, \nabla ^\sigma \bar{s} = 0 $, for $ \sigma $ and $ \bar{s} $ sections of $ C(P) $ and $ P/H $ respectively. We prove that the critical section of this problem are solutions of the Euler-Poincaré equations of the reduced problem. We also study the Hamilton-Cartan formulation of this Lagrange problem, where we find some common points with Pontryagin's approach to optimal control problems for $ \sigma $ as control variables and $ \bar{s} $ as dynamical variables. Finally, the theory is illustrated with the case of affine principal fiber bundles and its application to the modelisation of the molecular strands on a Lorentzian plane.

Keywords: Constraints, field theories, Lagrange multipliers, principal bundles, reduction, variational calculus.
Mathematics Subject Classification: Primary: 58E30; Secondary: 70S05, 53C05.
Citation: Marco Castrillón López, Pedro Luis García Pérez. The problem of Lagrange on principal bundles under a subgroup of symmetries. Journal of Geometric Mechanics, 2019, 11 (4) : 539-552. doi: 10.3934/jgm.2019026
References:
[1]

E. Bibbona, L. Fatibene and M. Francaviglia, Chetaev versus vakonomic prescriptions in constrained field theories with parametrized variational calculus, J. Math. Phys., 48 (2007), 032903, 14 pp. doi: 10.1063/1.2709848.  Google Scholar

[2]

A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24. Systems and Control. Springer-Verlag, New York, 2003. doi: 10.1007/b97376.  Google Scholar

[3]

M. Castrillón, P. L. García and C. Rodrigo, Euler-Poincaré reduction in principal fiber bundles and the problem of Lagrange, Diff. Geom. Appl., 25 (2007), 585-593.  doi: 10.1016/j.difgeo.2007.06.007.  Google Scholar

[4]

M. Castrillón, P. L. García and C. Rodrigo, Euler-Poincaré reduction in principal bundles by a subgroup of the structure group, J. Geom. Phys, 74 (2013), 352-369.  doi: 10.1016/j.geomphys.2013.08.008.  Google Scholar

[5]

M. Castrillón and P. L. García, Euler-Poincaré reduction by a subgroup of the symmetries as an optimal control problem, Geometry, Algebra and Applications: From Mechanics to Cryptography, Springer Proc. Math. Stat., Springer, [Cham], 161 (2016), 49-63.  doi: 10.1007/978-3-319-32085-4_5.  Google Scholar

[6]

D. C. P. Ellis, F. Gay-Balmaz, D. D. Holm, V. Putkaradze and T. S. Ratiu, Symmetry reduced dynamics of charged molecular strands, Arch. Ration. Mech. Anal., 197 (2010), 811-902.  doi: 10.1007/s00205-010-0305-y.  Google Scholar

[7]

P. L. García, The Poincaré-Cartan invariant in the calculus of variations, Symp. Math., Academic Press, London, 14 (1974), 219-249.   Google Scholar

[8]

P. L. García, Sobre la Regularidad en los Problemas de Lagrange y de Control Óptimo, El legado matemático de Juan Bautista Sancho Guimerá, Ediciones de la Universidad de Salamanca, 2015, 51–74. ISBN 978-84-9012-574-8. Google Scholar

[9]

P. L. García, A. García and C. Rodrigo, Cartan forms for first order constrained variational problems, J. Geom. Phys., 56 (2006), 571-610.  doi: 10.1016/j.geomphys.2005.04.002.  Google Scholar

[10]

J.-L. Koszul, Lectures on Fibers Bundles and Differential Geometry, Lect. in Math. No. 20, Tata Institute of Fundamental Research, Bombay, 1965.  Google Scholar

show all references

References:
[1]

E. Bibbona, L. Fatibene and M. Francaviglia, Chetaev versus vakonomic prescriptions in constrained field theories with parametrized variational calculus, J. Math. Phys., 48 (2007), 032903, 14 pp. doi: 10.1063/1.2709848.  Google Scholar

[2]

A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24. Systems and Control. Springer-Verlag, New York, 2003. doi: 10.1007/b97376.  Google Scholar

[3]

M. Castrillón, P. L. García and C. Rodrigo, Euler-Poincaré reduction in principal fiber bundles and the problem of Lagrange, Diff. Geom. Appl., 25 (2007), 585-593.  doi: 10.1016/j.difgeo.2007.06.007.  Google Scholar

[4]

M. Castrillón, P. L. García and C. Rodrigo, Euler-Poincaré reduction in principal bundles by a subgroup of the structure group, J. Geom. Phys, 74 (2013), 352-369.  doi: 10.1016/j.geomphys.2013.08.008.  Google Scholar

[5]

M. Castrillón and P. L. García, Euler-Poincaré reduction by a subgroup of the symmetries as an optimal control problem, Geometry, Algebra and Applications: From Mechanics to Cryptography, Springer Proc. Math. Stat., Springer, [Cham], 161 (2016), 49-63.  doi: 10.1007/978-3-319-32085-4_5.  Google Scholar

[6]

D. C. P. Ellis, F. Gay-Balmaz, D. D. Holm, V. Putkaradze and T. S. Ratiu, Symmetry reduced dynamics of charged molecular strands, Arch. Ration. Mech. Anal., 197 (2010), 811-902.  doi: 10.1007/s00205-010-0305-y.  Google Scholar

[7]

P. L. García, The Poincaré-Cartan invariant in the calculus of variations, Symp. Math., Academic Press, London, 14 (1974), 219-249.   Google Scholar

[8]

P. L. García, Sobre la Regularidad en los Problemas de Lagrange y de Control Óptimo, El legado matemático de Juan Bautista Sancho Guimerá, Ediciones de la Universidad de Salamanca, 2015, 51–74. ISBN 978-84-9012-574-8. Google Scholar

[9]

P. L. García, A. García and C. Rodrigo, Cartan forms for first order constrained variational problems, J. Geom. Phys., 56 (2006), 571-610.  doi: 10.1016/j.geomphys.2005.04.002.  Google Scholar

[10]

J.-L. Koszul, Lectures on Fibers Bundles and Differential Geometry, Lect. in Math. No. 20, Tata Institute of Fundamental Research, Bombay, 1965.  Google Scholar

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