# American Institute of Mathematical Sciences

December  2013, 5(4): 399-414. doi: 10.3934/jgm.2013.5.399

## Lagrange-Poincaré reduction in affine principal bundles

 1 ICMAT (CSIC-UAM-UC3M-UAM), Dpto. Geometría y Topología, Universidad Complutense de Madrid, 28040 Madrid, Spain 2 Dpto. Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain 3 IUFFyM-USAL and Real Academia de Ciencias, Plaza de la Merced 1-4, 37008 Salamanca

Received  June 2013 Revised  November 2013 Published  December 2013

Given an $H$-principal bundle $Q\to M$ and a (left) linear action of $H$ to a real vector space $V$, let $E\to M$ be the vector bundle associated to $Q$ and to the linear action, and $Q\times_M E$ the affine principal bundle with structure group the semidirect group $G = H Ⓢ V$. If $L v$ is a Lagrangian density defined on the 1-jet bundle $J^1(Q\times_M E)$ invariant by the subgroup $H \hookrightarrow H Ⓢ V$, the variational problem induced on $(J^1(Q\times_ME)) /H = C(Q)\times_M J^1E$, where $C(Q)$ is the bundle of connections in $Q$, is considered. We show that the reduced Lagrangian density $lv$ defines a variational problem on connections $\sigma \in \Gamma (C(Q))$ and on sections $e\in \Gamma(E)$, with constraint $\textrm{Curv }\sigma =0$, and set of admissible variations those induced on $\Gamma (C(Q))$ by the infinitesimal gauge transformations of $Q$ and on $\Gamma(E)$ by arbitrary vertical variations. The Lagrange-Poincaré equations for the critical reduced sections are obtained, as well as the reconstruction process to the unreduced problem. The Poincaré equation is interpreted as the reduction of the Noether conservation law corresponding to the $H$-symmetry of the Lagrangian density $L v$. We also study the reduced system as a Lagrange problem through a suitable choice of the Lagrange multipliers. This allows us to establish a Hamilton-Cartan formalism for this class of systems. Finally, we discuss the molecular strands, a motivating example of the theory.
Citation: Marco Castrillón López, Pablo M. Chacón, Pedro L. García. Lagrange-Poincaré reduction in affine principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 399-414. doi: 10.3934/jgm.2013.5.399
##### References:
 [1] M. Castrillón López, P. L. García Pérez and T. S. Ratiu, Euler-Poincaré reduction on principal bundles, Lett. Math. Phys., 58 (2001), 167-180. doi: 10.1023/A:1013303320765. [2] M. Castrillón López, P. L. García Pérez and C. Rodrigo, Euler-Poincaré reduction in principal fibre bundles and the problem of Lagrange, Differential Geom. Appl., 25 (2007), 585-593. doi: 10.1016/j.difgeo.2007.06.007. [3] M. Castrillón López, P. L. García Pérez 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. [4] M. Castrillón López and J. Muñoz Masqué, The geometry of the bundle of connections, Math. Z., 236 (2001), 797-811. [5] M. Castrillón López and T. S. Ratiu, Reduction in principal bundles: Covariant Lagrange-Poincaré equations, Comm. Math. Phys., 236 (2003), 223-250. doi: 10.1007/s00220-003-0797-5. [6] M. Castrillón López, T. S. Ratiu and S. Shkoller, Reduction in principal fiber bundles: Covariant Euler-Poincaré equations, Proc. Amer. Math. Soc., 128 (2000), 2155-2164. doi: 10.1090/S0002-9939-99-05304-6. [7] 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. [8] D. C. P. Ellis, F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Lagrange-Poincaré field equations, J. Geom. Phys., 61 (2011), 2120-2146. doi: 10.1016/j.geomphys.2011.06.007. [9] P. L. García, Gauge algebras, curvature and symplectic structure, J. Differential Geometry, 12 (1977), 209-227. [10] P. L. García, The Poincaré-Cartan invariant in the calculus of variations, in Symposia Mathematica, Vol. XIV (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973), Academic Press, London, 1974, 219-246. [11] 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. [12] H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier (Grenoble), 23 (1973), 203-267. doi: 10.5802/aif.451. [13] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. [14] S. Sternberg, Lectures on Differential Geometry, Second edition, Chelsea Publishing Co., New York, 1983.

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##### References:
 [1] M. Castrillón López, P. L. García Pérez and T. S. Ratiu, Euler-Poincaré reduction on principal bundles, Lett. Math. Phys., 58 (2001), 167-180. doi: 10.1023/A:1013303320765. [2] M. Castrillón López, P. L. García Pérez and C. Rodrigo, Euler-Poincaré reduction in principal fibre bundles and the problem of Lagrange, Differential Geom. Appl., 25 (2007), 585-593. doi: 10.1016/j.difgeo.2007.06.007. [3] M. Castrillón López, P. L. García Pérez 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. [4] M. Castrillón López and J. Muñoz Masqué, The geometry of the bundle of connections, Math. Z., 236 (2001), 797-811. [5] M. Castrillón López and T. S. Ratiu, Reduction in principal bundles: Covariant Lagrange-Poincaré equations, Comm. Math. Phys., 236 (2003), 223-250. doi: 10.1007/s00220-003-0797-5. [6] M. Castrillón López, T. S. Ratiu and S. Shkoller, Reduction in principal fiber bundles: Covariant Euler-Poincaré equations, Proc. Amer. Math. Soc., 128 (2000), 2155-2164. doi: 10.1090/S0002-9939-99-05304-6. [7] 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. [8] D. C. P. Ellis, F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Lagrange-Poincaré field equations, J. Geom. Phys., 61 (2011), 2120-2146. doi: 10.1016/j.geomphys.2011.06.007. [9] P. L. García, Gauge algebras, curvature and symplectic structure, J. Differential Geometry, 12 (1977), 209-227. [10] P. L. García, The Poincaré-Cartan invariant in the calculus of variations, in Symposia Mathematica, Vol. XIV (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973), Academic Press, London, 1974, 219-246. [11] 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. [12] H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier (Grenoble), 23 (1973), 203-267. doi: 10.5802/aif.451. [13] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. [14] S. Sternberg, Lectures on Differential Geometry, Second edition, Chelsea Publishing Co., New York, 1983.
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