×

Euler-Poincaré reduction in principal bundles by a subgroup of the structure group. (English) Zbl 1282.58005

Summary: Given a Lagrangian density \(L\mathbf{v}\) defined on the 1-jet bundle \(J^1P\) of a principal \(G\)-bundle \(\pi \colon P \to M\) invariant with respect to a subgroup \(H\) of \(G\), the reduction of the variational problem defined by \(L\mathbf{v}\) to \((J^1P)/H = C \times_M(P/H)\), where \(C\) is the bundle of connections in \(P\), is studied. It is shown that the reduced Lagrangian density \(l\mathbf{v}\) defines a zero order variational problem on connections \(\sigma\) and \(H\)-structures \(\bar{s}\) of \(P\) with non-holonomic constraints \(\text{Curv}\sigma = 0\) and \(\nabla^\sigma\bar{s} = 0\) and set of admissible variations those induced by the infinitesimal gauge transformations in \(C\) and \(P/H\). The Euler-Poincaré equations for critical reduced sections are obtained as well as the reconstruction process to the unreduced problem. The corresponding conservation laws and their relationship with the Noether theory are also analyzed. Finally, some instances are studied: the heavy top and affine principal bundles, the main application of which is used for molecular strands.

MSC:

58A15 Exterior differential systems (Cartan theory)
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Castrillón López, M.; Ratiu, T. S.; Shkoller, S., Reduction in principal fiber bundles: covariant Euler-Poincaré equations, Proc. Amer. Math. Soc., 128, 7, 2155-2164 (2000) · Zbl 0967.53019
[2] Castrillón López, M.; García Pérez, P.; Ratiu, T., Euler-Poincaré reduction on principal bundles, Lett. Math. Phys., 58, 167-180 (2001) · Zbl 1020.58018
[3] Castrillón López, M.; García Pérez, P.; Rodrigo, C., Euler-Poincaré reduction in principal fibre bundles and the problem of Lagrange, Differential Geom. Appl., 25, 6, 585-593 (2007) · Zbl 1132.58001
[4] Castrillón López, M.; Ratiu, T. S., Reduction in principal bundles: covariant Lagrange-Poincaré equations, Comm. Math. Phys., 236, 2, 223-250 (2003) · Zbl 1037.53056
[5] Ellis, D. C.P.; Gay-Balmaz, F.; Holm, D. D.; Ratiu, T. S., Lagrange-Poincaré field equations, J. Geom. Phys., 61, 11, 2120-2146 (2011) · Zbl 1253.70031
[6] Ellis, D. C.P.; Gay-Balmaz, F.; Holm, D. D.; Putkaradze, V.; Ratiu, T. S., Symmetry reduced dynamics of charged molecular strands, Arch. Ration. Mech. Anal., 197, 3, 811-902 (2010) · Zbl 1333.70029
[7] Castrillón López, M.; Muñoz Masqué, J., The geometry of the bundle of connections, Math. Z., 236, 4, 797-811 (2001) · Zbl 0977.53020
[8] García, P. L., Gauge algebras, curvature and symplectic structure, J. Differential Geom., 12, 209-227 (1977) · Zbl 0404.53033
[9] Kobayashi, S.; Nomizu, K., Foundations of Differential Geometry, Vol. I (1963), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York, Vol. II, 1969 · Zbl 0119.37502
[10] García, P. L., The Poincaré-Cartan invariant in the calculus of variations, (Symposia Mathematica, vol. XIV (1974), Academic Press: Academic Press London), 219-246
[11] Goldsmith, H.; Sternberg, S., The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier (Grenoble), 23, 1, 203-267 (1973) · Zbl 0243.49011
[12] Varadarajan, V. S., Lie Groups, Lie Algebras, and their Representations (1984), Springer-Verlag New York Inc. · Zbl 0955.22500
[13] Arnold, V. I., (Dynamical Systems III. Dynamical Systems III, Encyclopedia of Mathematics, vol. 3 (1988), Springer-Verlag)
[14] Marsden, J. E.; Ratiu, T. S., (Introduction to Mechanics and Symmetry. Introduction to Mechanics and Symmetry, Text in Applied Mathematics, vol. 17 (1999), Springer-Verlag) · Zbl 0933.70003
[15] Holm, D.; Marsden, J. E.; Ratiu, T. S., The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137, 1-81 (1998) · Zbl 0951.37020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.