# American Institute of Mathematical Sciences

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

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

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.

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. [2] A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24. Systems and Control. Springer-Verlag, New York, 2003. doi: 10.1007/b97376. [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. [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. [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. [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. [7] P. L. García, The Poincaré-Cartan invariant in the calculus of variations, Symp. Math., Academic Press, London, 14 (1974), 219-249. [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. [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. [10] J.-L. Koszul, Lectures on Fibers Bundles and Differential Geometry, Lect. in Math. No. 20, Tata Institute of Fundamental Research, Bombay, 1965.

show all references

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

##### 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. [2] A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24. Systems and Control. Springer-Verlag, New York, 2003. doi: 10.1007/b97376. [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. [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. [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. [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. [7] P. L. García, The Poincaré-Cartan invariant in the calculus of variations, Symp. Math., Academic Press, London, 14 (1974), 219-249. [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. [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. [10] J.-L. Koszul, Lectures on Fibers Bundles and Differential Geometry, Lect. in Math. No. 20, Tata Institute of Fundamental Research, Bombay, 1965.
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