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.  doi: 10.1023/A:1013303320765.  Google Scholar

[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.  doi: 10.1016/j.difgeo.2007.06.007.  Google Scholar

[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.  doi: 10.1016/j.geomphys.2013.08.008.  Google Scholar

[4]

M. Castrillón López and J. Muñoz Masqué, The geometry of the bundle of connections,, Math. Z., 236 (2001), 797.   Google Scholar

[5]

M. Castrillón López and T. S. Ratiu, Reduction in principal bundles: Covariant Lagrange-Poincaré equations,, Comm. Math. Phys., 236 (2003), 223.  doi: 10.1007/s00220-003-0797-5.  Google Scholar

[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.  doi: 10.1090/S0002-9939-99-05304-6.  Google Scholar

[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.  doi: 10.1007/s00205-010-0305-y.  Google Scholar

[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.  doi: 10.1016/j.geomphys.2011.06.007.  Google Scholar

[9]

P. L. García, Gauge algebras, curvature and symplectic structure,, J. Differential Geometry, 12 (1977), 209.   Google Scholar

[10]

P. L. García, The Poincaré-Cartan invariant in the calculus of variations,, in Symposia Mathematica, (1973), 219.   Google Scholar

[11]

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

[12]

H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations,, Ann. Inst. Fourier (Grenoble), 23 (1973), 203.  doi: 10.5802/aif.451.  Google Scholar

[13]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I,, Interscience Publishers, (1963).   Google Scholar

[14]

S. Sternberg, Lectures on Differential Geometry,, Second edition, (1983).   Google Scholar

show all references

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.  doi: 10.1023/A:1013303320765.  Google Scholar

[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.  doi: 10.1016/j.difgeo.2007.06.007.  Google Scholar

[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.  doi: 10.1016/j.geomphys.2013.08.008.  Google Scholar

[4]

M. Castrillón López and J. Muñoz Masqué, The geometry of the bundle of connections,, Math. Z., 236 (2001), 797.   Google Scholar

[5]

M. Castrillón López and T. S. Ratiu, Reduction in principal bundles: Covariant Lagrange-Poincaré equations,, Comm. Math. Phys., 236 (2003), 223.  doi: 10.1007/s00220-003-0797-5.  Google Scholar

[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.  doi: 10.1090/S0002-9939-99-05304-6.  Google Scholar

[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.  doi: 10.1007/s00205-010-0305-y.  Google Scholar

[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.  doi: 10.1016/j.geomphys.2011.06.007.  Google Scholar

[9]

P. L. García, Gauge algebras, curvature and symplectic structure,, J. Differential Geometry, 12 (1977), 209.   Google Scholar

[10]

P. L. García, The Poincaré-Cartan invariant in the calculus of variations,, in Symposia Mathematica, (1973), 219.   Google Scholar

[11]

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

[12]

H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations,, Ann. Inst. Fourier (Grenoble), 23 (1973), 203.  doi: 10.5802/aif.451.  Google Scholar

[13]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I,, Interscience Publishers, (1963).   Google Scholar

[14]

S. Sternberg, Lectures on Differential Geometry,, Second edition, (1983).   Google Scholar

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