RT Journal Article
T1 Euler-Poincaré reduction on principal bundles
A1 Castrillón López, Marco
A1 García Pérez, P.L
A1 Ratiu, T.S.
AB Let G be a Lie group and let L:TG→R be a Lagrangian invariant under the natural action of G on its tangent bundle. Then L induces a function l:(TG)/G≅g→R called the reduced Lagrangian, g being the Lie algebra of G. As is well known, the Euler-Lagrange equations defined by L for curves on G are equivalent to a new kind of equation for l for the reduced curves in the Lie algebra g. These equations are known as the Euler-Poincaré equations. In the paper under review, the authors extend the idea of the Euler-Poincaré reduction to a Lagrangian L:J1P→R defined on the first jet bundle of an arbitrary principal bundle π:P→M with structure group G. The Lagrangian is assumed to be invariant under the natural action of G on J1P. Let l:(J1P)/G→R be the reduced Lagrangian. It is known that the quotient manifold (J1P)/G can be identified with the bundle of connections of π:P→M. The reduced variational problem has a nice geometrical interpretation in terms of connections. The authors study the compatibility conditions needed for reconstruction. In this framework the Euler-Poincaré equations do not suffice to reconstruct the Euler-Lagrange equations. Some extra conditions must be imposed, namely, the vanishing of the curvature of the critical sections. In the case of matrix groups this result has already been obtained [M. Castrillón López, T. S. Ratiu and S. Shkoller, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2155–2164;]. In this paper the authors give a proof for general Lie groups. Moreover, they point out several facts concerning the reduced variational problem: its relation with the variational calculus with constraints, Noether's theorem for reduced symmetries, and the second variation formula.
PB Kluwer Academic
SN 0377-9017
YR 2001
FD 2001-11
LK https://hdl.handle.net/20.500.14352/58890
UL https://hdl.handle.net/20.500.14352/58890
LA eng
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NO DGESIC (Spain)  through fundin
NO J. Castilla León (Spain)
NO European Commission and the Swiss Federal Government
NO Research Training Network Mechanics and Symmetry in Europe (MASIE)
NO Swiss National Science Foundation.
DS Docta Complutense
RD 3 may 2024