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Euler-Poincare reduction in principal fibre bundles and the problem of Lagrange

Castrillón López, Marco and Rodrigo, César and Garcia, Pedro L. (2007) Euler-Poincare reduction in principal fibre bundles and the problem of Lagrange. Differential Geometry and Its Applications, 25 (6). pp. 585-593. ISSN 0926-2245

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Abstract

We compare Euler–Poincaré reduction in principal fibre bundles, as a constrained variational problem on the connections of
this fibre bundle and constraint defined by the vanishing of the curvature of the connection, with the corresponding problem of
Lagrange. Under certain cohomological condition we prove the equality of the sets of critical sections of both problems with the
one obtained by application of the Lagrange multiplier rule. We compute the corresponding Cartan form and characterise critical
sections as the set of holonomic solutions of the Cartan equation and, in particular, under a certain regularity condition for the
problem, we prove the holonomy of any solution of this equation.

Item Type:Article
Uncontrolled Keywords:Variational problems; problem of Lagrange; Lagrange multipliers; Euler-Poincare equations; Mathematics, Applied; Mathematics
Subjects:Sciences > Mathematics > Differential geometry
ID Code:14875
References:

M. Castrillón, T. Ratiu, S. Shkoller, Reduction in principal fiber bundles: Covariant Euler–Poincaré equations, Proc. Amer. Math. Soc. 128 (2000).

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P.L. García, Connections and 1-jet fiber bundles, Rend. Sem. Mat. Univ. Padova 47 (1972) 227–242.

P.L. García, J. Muñoz, Higher order regular variational problems, in: Symplectic geometry and mathematical physics, Aix-en-Provence, 1990, in: Progr. Math., vol. 99, Birkhäuser, Boston, 1991, pp. 136–159.

P.L. García, A. García, C. Rodrigo, Cartan forms for first order constrained variational problems, J. Geom. Phys. 56 (2006) 571–610.

Deposited On:19 Apr 2012 08:32
Last Modified:06 Feb 2014 10:11

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