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